Petri Nets 2000 · Petri Nets 2000 21st International Conference on Application and Theory of Petri Nets Aarhus, Denmark, June 26-30, 2000 Introductory Tutorial Petri Nets

Petri Nets 200021st International Conference onApplication and Theory of Petri Nets

Aarhus, Denmark, June 26-30, 2000

Introductory TutorialPetri Nets

Organised by

Gianfranco Balbo

Jorg DeselKurt Jensen

Wolfgang ReisigGrzegorz Rozenberg

Manuel Silva

June 2000

DEPARTMENT OF COMPUTER SCIENCEUNIVERSITY OF AARHUS

Ny Munkegade, Bldg. 540DK-8000 Aarhus C, Denmark

Table of Contents

Wolfgang ReisigAn Informal Introduction to Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . 1

Grzegorz RozenbergElementary Net Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Jorg DeselPlace/Transition Nets I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Kurt JensenColoured Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Grzegorz RozenbergBehaviour of Elementary Net Systems . . . . . . . . . . . . . . . . . . . . . . . . . 161

Manuel SilvaPlace/Transition Nets II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Gianfranco BalboAn Introduction to Generalised Stochastic Petri Nets . . . . . . . . . . . . . . . . . 217

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1Introductory Tutorial

Place/Transition-Nets I

Jorg Desel, Catholic University in Eichstatt

I. Introduction to place/transition-nets

II. Basic analysis techniques

2I. Introduction to An example

place/transition nets Different features of place/transition-nets

Place/transition-nets vs en-systems

Formal definitions

Place/transition-nets

Occurrence sequences and reachability

Marking graphs

Behavioral properties

Deadlock-freedom and liveness

Boundedness and 1–safety

Reversibility

Capacities and complements

Weak and strong capacities

Weak and strong complements

Inhibitor arcs85

eady for insertion

holding coin

dispense item

nsert coin

ccept coin

reject coin

ready to dispense

eady for insertion

holding coin

ispenseitem

nsert coin

ccept coin

reject coin

eady to dispense

3An example: a vending machine

Control structure of a vending machine

an en-system its behaviour

item storage

request for refill

ready for insertion

holding coin

refilldispense item

insert coin

accept coin

reject coin

ready to dispense

ispenseitem

insert coin

accept coin

reject coin

insert coin

accept coin

reject coin

refill

refill

refill

4Adding concurrency: a vending machine with capacity 1 ...

... and its behaviour

86

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

ic

rc

ic ic ic ic

rc rc rc rc

refill refill refill

refillrefillrefillrefill

refill

ac ac ac acac

di di di di

refill refill refill refill

5Adding bounded storages: a vending machine with capacity 4 ...


ready for insertion

holding coindispense item

insert coin

accept coin

reject coin

ready to dispense

coins in machine

dispense item

insert coin

accept coin

reject coin

insert coin

accept coin

reject coin

dispense item

0 coins in machine 1 coin in machine 2 coins in machine

6Adding unbounded counters: the control part with a counter ...


87

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

7Adding arc weights: the vending machine selling pairs ...

... or storing pairs

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

tem storage

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

eject coin

eady to dispense

= 4

8Adding limited capacities: replacing the place ”request for refill” ...

... by a capacity restriction

88

9Marked place/transition-nets generalize en-systems

Each contact-free en-system is a 1-safe marked place/transition-net

Terminology:

en-system marked p/t-net

condition �! place

event �! transition

case / state �! marking

c � conditions m: places ! f0; 1gsequential case graph �! marking graph

(reachability graph, state graph)

10Formal definition of marked place/transition-nets

A marked place/transition-net (p/t-net) is a tuple (S; T; F; k; w;m0) where

(S; T; F ) is a net with

S – set of places (Stellen), nonempty, finite (often P is used)

T – set of transitions , nonempty, finite

F � (S � T ) [ (T � S) – flow relation

k :S ! f1; 2; 3; . . .g [ f1g – partial capacity restriction (default: 1)

w :F ! f1; 2; 3; . . .g – weight function (default: 1)

m0:S ! f0; 1; 2; . . .g – a marking satisfying

8s 2 S : k(s) = 1 _ m0(s) � k(s)

(initial marking )

89

2

2

32

2

3 k=4

11The occurrence rule

A transition t is enabled at a marking m if

every place s 2 �t satisfies m(s) � w(s; t) and

every place s 2 t� satisfies m(s) + w(t; s) � k(s)

The occurrence of t leads to the successor marking m0, defined by

m0(s) =

8>>>>><>>>>>:

m(s) if s =2 �t and s =2 t�

m(s)� w(s; t) if s 2 �t and s =2 t�

m(s) + w(t; s) if s =2 �t and s 2 t�

m(s)� w(s; t) + w(t; s) if s 2 �t and s 2 t�

Notation: mt�! m0 (m[tim0)

12Occurrence sequences and reachability

A finite sequence � = t1 t2 . . . tn of transitions is a

finite occurrence sequence leading from m0 to mn if

m0t1�! m1

t2�! � � � tn�! mn

A marking m is reachable (from m0) if

there is an occurrence sequence leading from m0 to m

Notation: [m0i is the set of all reachable markings

An infinite sequence � = t1 t2 t3 . . . is an

infinite occurrence sequence enabled at m0 if

m0t1�! m1

t2�! m2t3�! � � �

90

(1,1,0)t1 t2

s1

s2

s3 (0,2,0)

t2 (1,0,1)

t1

t2 (0,1,1)

t1

(0,0,2)

t2

t3

2 t3

13Marking graphs

The marking graph of a marked p/t-net is an edge-labeled graph with initial vertex

initial vertex – initial marking m0 (denoted !�)

vertices – set of reachable markings [m0ilabeled edges – set of triples (m; t;m0) such that m

t�! m0

Example:

Lemma Each occurrence sequence corresponds to the

labels of a directed path of the marking graph

starting with the initial vertex,

and vice versa.

14Behavioral properties of marked p/t-nets

A marked p/t-net is

terminating – if there is no infinite occurrence sequence

deadlock-free – if each reachable marking enables a transition

live – if each reachable marking enables

an occurrence sequence containing all transitions

bounded – if, for each place s, there is a bound b(s)

such that m(s) � b(s) for every reachable marking m

1–safe – if b(s) = 1 is a bound for each place s

reversible – if m0 is reachable from each other reachable marking

Example The vending machines are deadlock-free and live.

Some are 1–safe, some are bounded, some are unbounded.

The bounded vending machines are reversible.91

t1t2

t2

t3

t4

t1

t4

t3

t4 t2

15A marked p/t-net which is not deadlock-free and its marking graph

Proposition A marked p/t-net is deadlock-free if and only if

its marking graph has no vertex without successor

Proposition No deadlock-free marked p/t-net is terminating

(but the converse does not necessarily hold)

t2 t4

t1 t5

t3t2

t1

t3

t4

t5

4

t5

2 t4

1 5

3

16A deadlock-free marked p/t-net which is not live

Proposition Every live marked p/t-net is deadlock-free

(this does not hold for nets without transitions)

Proposition A marked p/t-net is live if and only if

at no reachable marking a transition is dead

(cannot become enabled again)

Example Some transitions are dead at a reachable marking

92

17Proposition A marked p/t-net is bounded if and only if

its set of reachable markings is finite

(its marking graph is finite)

Proof

(() The maximal number of tokens on a place can be taken as its bound.

()) If a place s is bounded by b(s) then it can be in at most b(s) + 1 different states, vic.

m(s) = 0;m(s) = 1; . . . ;m(s) = b(s):

So the number of reachable markings does not exceed

(b(s1) + 1) � (b(s2) + 1) � � � (b(sn) + 1)

where fs1; s2; . . . ; sng is the (finite !) set of places

Corollary A 1–safe marked p/t-net with n places has

at most 2n reachable markings

t2 t4

t1 t5

t3t2

t1

t3

t4

t5

t1

t2

t3

t4

t5

t3

t3t4

t4t5

t2

t1

18Proposition A marked p/t-net is reversible if and only if

its marking graph is strongly connected

Example a 1–safe non-live marked p/t-net which is not reversible

Example an unbounded marked p/t-net which is not reversible

Example a live and 1–safe marked p/t-net which is not reversible

93

tem storage

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

eject coin

eady to dispense

= 4

tem storage

equest for refill

eady for insertion

holding coin

efillispense

item

nsert coin

accept coin

reject coin

eady to dispense

19Substituting capacities ...

... by complement places

k = 3

20Weak capacities

. . . guarantee bounds of places

weak enabling condition:

a transition t is enabled at a marking m if


every place s 2 t� n �t satisfies m(s) + w(t; s) � k(s) and

every place s 2 t� \ �t satisfies m(s)� w(s; t) + w(t; s) � k(s)

Proposition If k(s) is finite then s is k(s)-bounded

Replacing a weak capacity restrictiction by a weak complement

94

= 3

21Strong capacities

. . . generalize contact of en-systems

strong enabling condition:

a transition t is enabled at a marking m if


every place s 2 t� satisfies m(s) + w(t; s) � k(s)

Proposition each en-system is equivalent to a

marked p/t-net without arc weights and

with the strong capacity restriction k(s) = 1 for every place s

Replacing a strong capacity restrictiction by a strong complement

k = 3 3 3

22Inhibitor arcs for null tests

inhibitor enabling condition: If (s; t) is an inhibitor arc then

t is only enabled at a marking m if m(s) = 0

Replacing an inhibitor arc at a bounded place by a weak complement

95

23II. Basic Linear-algebraic techniques

analysis The marking equation

techniques Place invariants

Transition invariants

Structural techniques

Siphons

Traps

The siphon/trap property

Restricted net classes

State machines

Marked graphs

Free-choice nets

Causal Semantics

Occurrence nets

Process nets

1

s2

s3

s4t1 t2

3

4

t5

5

24Linear-algebraic representation of markings and transitions

vector representation of the marking m0: ~m0 = (4; 0; 0; 0; 1)

vector representation of the transition t2: ~t2 = (�1; 1; 1; 0;�1)

m0t2�! m1 ) ~m0 + ~t2 = ~m1 = (3; 1; 1; 0; 0)

96

1

s2

s3

s4t1 t2

3

4

t5

5

25Matrix representation of a net

incidence matrix of the net:

��N ��=

~t1 ~t2 ~t3 ~t4 ~t5

~s1 1 �1 0 0 0

~s2 �1 1 0 0 0

~s3 0 1 �1 0 1

~s4 0 0 1 �1 �1

~s5 0 �1 0 1 0

26The marking equation

m0

t2 t3 t5 t1 t3��! m ) ~m0 + ~t2 + ~t3 + ~t5 + ~t1 + ~t3 = ~m

~m0 + (1 � ~t1) + (1 � ~t2) + (2 � ~t3) + (0 � ~t4) + (1 � ~t5) = ~m

~m0 +��N �� (1; 1; 2; 0; 1)| {z }

Parikh vector of t2t3t5t1t3

= ~m

The Marking Equation If m0��! m and

P(�) denotes the Parikh vector of � then

~m0 +��N �� P(�) = ~m

. . . yields a necessary condition for reachability of a marking:

A marking m is only reachable from m0 if

~m0 +��N �� ~x = ~m has a solution for ~x in IN

�.

97

t1

2

3

t4

1

2

3

4

5

5

27Example: a live and 1–safe marked p/t-net

reachable markings solutions to the marking equation

(1; 0; 0; 0; 0) (0; 0; 0; 0; 0); (1; 0; 1; 0; 1); (0; 1; 0; 1; 1); . . .

(0; 1; 0; 0; 1) (1; 0; 0; 0; 0); . . .

(0; 0; 1; 1; 0) (0; 1; 0; 0; 0); . . .

(0; 0; 0; 1; 1) (1; 0; 1; 0; 0); (0; 1; 0; 1; 0); . . .

non-reachable marking solutions to the marking equation

(0; 1; 1; 0; 0) (1; 1; 0; 0; 1) . . .

Consequence: solubility of the marking equation is not sufficient for reachability

s1 s2

s3

t1

t2

s4

t3

t4

s5

process 1 process 2

28Place invariants

Example: mutual exclusion

Every reachable marking m satisfies m(s2) + m(s4) � 1

1) m(s2) + m(s3) + m(s4) = 1 holds initially

2) m(s2) + m(s3) + m(s4) = 1 is stable ! will be shown by a place invariant

3) m(s2) + m(s3) + m(s4) = 1 ) m(s2) + m(s4) � 198

29Place invariants

Three equivalent definitions:

A place invariant of a net N is a vector~i satisfying

(1)Xs2�t

~is =Xs2t�

~is for every transition t of N

(2)~i � ~t = 0 for every transition t of N

(3)~i � ��N ��= (0; 0; . . . ; 0)

The token conservation law for a place invariant ~i:

If m is reachable from m0 then~i � ~m0 =~i � ~m

Proof: m0��! m ) ~m0 +

��N �� P [�] = ~m

) ~i � ~m0 + ~i � ��N ��| {z }=(0;...;0)

�P [�] =~i � ~m

) ~i � ~m0 =~i � ~m

s1 s2

s3

t1

t2

s4

t3

t4

s5

process 1 process 2

30Proving stability

The number of tokens on fs2; s3; s4g is not changed

by transition occurrences.

)~i = (0; 1; 1; 1; 0) is a place invariant.

~i � ~m0 = 1 implies~i � ~m = 1 for each reachable marking M .

) m(s2) + m(s3) + m(s4) = 1 is stable.

99

s1 s2

s3

t1

t2

s4

t3

t4

s5

process 1 process 2

31Further place invariants

(0; 1; 1; 1; 0) mutual exclusion

(0; 1; 1; 0;�1) m(s2) + m(s3) = m(s5)

if s2 is marked then s5 is marked

(1; 1; 0; 0; 0) m(s1) + m(s2) = 1

m(s1);m(s2) � 1, the places s1 and s2 are bounded

1 2

3

4

1

2

t3

t4

s5

32A necessary condition for liveness

Proposition: In a live marked p/t-net without isolated places,

each place invariant~i without negative entries and

with some positive entry~is satisfies~i � ~m0 > 0.

Proof: otherwise transitions in �s [ s� are dead.

Examples: place invariants (1; 1; 0; 0; 0), (0; 0; 0; 1; 1), (0; 1; 1; 1; 0)

100

1 2

3

4

1

2

t3

t4

s5

33A sufficient condition for boundedness:

Proposition: Each marked p/t-net with a place invariant~i satisfying

~is > 0 for each place s is bounded.

Proof: m is reachable ) ~i � ~m =~i � ~m0.

) ~is � ~ms �~i � ~m =~i � ~m0.

) m(s) = ~ms �~i � ~m0

~is

Example: place invariant (1; 2; 1; 2; 1)

t1 t2t3

t4 t5

s1

s2

s3

s4

s5

s6

s7

s8

34Place invariants and the marking equation

Proposition There is a place invariant~i satisfying~i � ~m0 6=~i � ~m if and only if

~m0 +��N �� ~x = ~m has no rational-valued solution for ~x.

Example:

~m0 +��N �� (1; 0; 1; 12 ;

12 ) = ~m = (1; 0; 1; 0; 1; 1; 0; 0)

) no place invariant proves the non-reachability of m.

But the marking equation has no solution in IN�

! modulo place invariants101

1 2

3

4

1

2

t3

t4

s5

35Transition invariants

A transition invariant of a net N is a vector ~j satisfying��N ��~j = (0; 0; . . . ; 0)

Example:

Transition invariants: (1; 1; 0; 0), (0; 0; 1; 1), (2; 2; 1; 1)

Proposition Let m0��! m be an occurrence sequence.

m0 = m if and only if P [�] is a transition invariant

Proof: follows immediately from ~m0 +��N �� P [�] = ~m

36A necessary condition for liveness and boundedness

Proposition: Each live and bounded marked p/t-net

has a transition invariant ~j satisfying

~j(t) > 0 for each transition t.

Proof: By liveness, there exist occurrence sequences

m0�1�! m1

�2�! m2�3�! � � �

such that all transitions occur in every �i.

By boundedness, mi = mj for some i < j.

) mi�i+1�! � � � �j�! mj = mi.

) ~j = P [�i+1 � � ��j ] is a suitable transition invariant.

102

a siphon {s1,s2}

a trap{s3, s4}

s1 s2 s3 s4

37Structural Techniques

A siphon is a set of places which, once unmarked, never gains a token again

S is a siphon if �S � S�, i.e. if t� \ S 6= ; implies �

t \ S 6= ;.

A trap is a set of places which, once marked, never looses all tokens

S is a trap if S� � �S, i.e. if �t \ S 6= ; implies t� \ S 6= ;.

If a marking m satisfies m(s1) = m(s2) = 0 then so do all follower markings.

If a marking m satisfies m(s3) + m(s4) > 0 then so do all follower markings.

1

2

3

t4

1

2

3

s4

5

t5

38Example for the use of a trap

fs1; s4; s5g is an initially marked trap

) the marking (0; 1; 1; 0; 0) is not reachable.

103

39Siphons and traps, liveness and deadlock-freedom

Proposition: In a live marked p/t-net without isolated places,

each nonempty siphon contains an initially marked place

Proof: otherwise, for each place s of the siphon, all transitions in �s [ s� are dead.

Proposition: Assume a marked p/t-net with some transition,

without capacity restrictions and arc weights.

If each nonempty siphon includes an initially marked trap

then the marked p/t-net is deadlock-free

Proof: the set of unmarked places at a dead marking is a nonempty siphon.

This siphon contains no marked trap.

) It contains no initially marked trap.

40Restricted net classes

State machines are marked p/t-nets without branched transitions,

i.e. j�tj = jt�j = 1 for each transition,

without arc weights and

without capacity restrictions.

Example

Proposition Each marked state machine is bounded.

Proposition A marked state machine is live if and only if

it is strongly connected and some place is initially marked.

104

41Marked graphs are marked p/t-nets without branched places,

i.e. j�sj = js�j = 1 for each place,

without arc weights and

without capacity restrictions.

Example

Proposition A marked graph is live if and only if

each cycle carries a token initially.

Proposition It is moreover 1–safe if and only if

each place belongs to a cycle with exactly one token.

s1

s2

s3

s4

s5

42Free-choice nets are marked p/t-nets

without arc weights and capacity restrictions satisfying

(s; t) 2 F ) �t� s

� � F for each place s and transition t

If then

Proposition A free-choice net without isolated places is live if and only if

each nonempty siphon includes an initially marked trap.

Example

fs1; s2; s5g is a siphon which includes no nonempty trap

) this free-choice net is not live.105

produce send receive consume

produced

sent

full

consumed

received

ent

onsumed

end

roduce

roduced ent

ull ullroduce

roduced send sent

eceive

eceived

consume

onsumed

receive

eceived

43Causal semantics of marked p/t-nets

Example: a producer / consumer system

a causal run of the producer /consumer system

44Causal runs

A causal run of a marked p/t-net is given by a labeled Petri net (B;E;K)

Interpretation of causal runs

net element name symbol interpretation

places conditions B tokens on system places

transitions events E system transition occurrences

arcs causal relation K flow of tokens

106

45Occurrence nets

An occurrence net is a net (B;E;K) with the following properties

it has no cycles (i.e. K+ is a partial order � )

it has no branched places, i.e.

j�bj; jb�j � 1 for each condition b

events have finite fan-in and fan-out, i.e.�e and e� are finite sets for each event e

it has neither input nor output-events, i.e.

j�ej; je�j � 1 for each event e

no node has infinitely many predecessors, i.e.

the set fx 2 (B [ E) j x � yg is finite for each node y

46Process nets of marked p/t-nets represent causal runs

Assume a marked p/t-net (S; T; F; k; w;m0) without capacity restrictions

An occurrence net (B;E;K) together with

labels �: (B [ E) ! (S [ T ) is a process net of N if

sorts of nodes are respected by �, i.e.

�(B) � S and �(E) � T

m0 agrees with min(B), i.e.

m0(s) = jfb 2 B j �b = ; and �(b) = sgj for every place s 2 S

transition vicinities are preserved, i.e.

�(�e) = �(�(e)), jfb 2 �e j �(b) = sgj = w(s; �(e)) for each event e

�(e�) = (�(e))�, jfb 2 e� j �(b) = sgj = w(�(e); s) for each event e

107

d a

e b

f

g

c h

d a

e b

f

g

c h

f

d a

e b

f

g

c hf

47Occurrence sequences versus process nets

occurrence provide total orders of events that respect causality

sequences but add arbitrary interleavings of independent events.

Information about causal relationships can get lost.

process nets provide partial orders reflecting causality.

Example:

a b c

b a c

a c b

b c a

maximal occurrence sequences maximal process nets

s1 s3 s4

s2 s6

s5

t1 t2

t3

t5

t4

b8

b3

b1 e1

b4

b5 e2 b6

e3

b2

e4

b14

b7

b9

e5

b10

b11 e6 b12

e7

b13s3 t3

s4

s5 t5 s6 t2

s1

s3 t3

s4

s5 t5 s6 t2 s3

s2s2 t1

s1

s2

e8

b15

b16 e9 b17t3

s4

s5 t4 s3

b8b3

b1 e1

b4

b5 e2 b6

e3

b2

e4

b14

b7

b9

e5

b10

b11 e6 b12

e7

b13s3 t3

s4

s5 t5 s6 t2

s1

s3 t3

s4

s5 t5 s6 t2 s3

s2s2 t1 s1 s2

e8

b15

b16 e9 b17t3

s4

s5 t4 s3

48Two process nets corresponding to t1 t3 t5 t2 t3 t5 t2 t3 t4

108

s1 s3 s4

s2 s6

s5

t1 t2

t3

t5

t4

b8

b3

b1 e1

b4

b5 e2 b6

e3

b2

e4

b14

b7

b9

e5

b10

b11 e6 b12

e7

b13s3 t3

s4

s5 t5 s6 t2

s1

s3 t3

s4

s5 t5 s6 t2 s3

s2s2 t1

s1

s2

e8

b15

b16 e9 b17t3

s4

s5 t4 s3

b2

b8

b1

b3

e1

b4

b5 e2 b6

e3

b7s3 t3

s4

s5 t5 s6 t2 s3

s2

s1

s2

e4

b9

b10 e6 b11t3

s4

s5 t4 s3

b12e5t1 s1

b17

e7

b13

b14 e8 b15

e9

b16t3

s4

s5 t5 s6 t2 s3

s2

49Two process nets without a common occurrence sequence

109

110

Col

oure

d P

etri

Net

s

Kur

t Jen

sen

Com

pute

r S

cien

ce D

epar

tmen

tU

nive

rsity

of A

arhu

s

Ny

Mun

kega

de, B

uild

ing

540

DK

-800

0 A

arhu

s C

, Den

mar

k

Pho

ne:

+45

89

42 3

2 34

Tel

efax

:+

45 8

9 42

32

55E

-mai

l: kj

ense

n@da

imi.a

u.dk

UR

L: h

ttp://

ww

w.d

aim

i.au.

dk/~

kjen

sen

TO

OL

S •

edi

ting

• s

imul

atio

n •

veri

fica

tion

TH

EO

RY

• m

odel

s•

basi

c co

ncep

ts•

veri

fica

tion

met

hods

PR

AC

TIC

AL

USE

• s

peci

fica

tion

• in

vest

igat

ion

• v

erif

icat

ion

•im

plem

enta

tion

Col

oure

d P

etri

Net

s 2

Par

t 1: I

ntro

duct

ion

to C

P-n

ets

An

ordi

nary

Pet

ri ne

t (P

T-n

et)

has no

typ

es a

nd no

mo

du

les:

•O

nly

one

kind

of t

oken

s an

d th

e ne

t is

flat.

With

Col

oure

d P

etri

Net

s (C

P-n

ets)

it is

pos

sibl

e to

use d

ata

typ

es a

nd c

ompl

ex da

ta m

an

ipu

latio

n:

•E

ach

toke

n ha

s at

tach

ed a

dat

a va

lue

calle

d th

eto

ken

co

lou

r.

•T

he to

ken

colo

urs

can

be

inve

stig

ate

d and

mo

difi

ed b

y th

e oc

curr

ing

tran

sitio

ns.

With

CP

-net

s it

is p

ossi

ble

to m

ake hie

rarc

hic

al

desc

riptio

ns:

•A

larg

e m

odel

can

be

obta

ined

by

com

bin

ing a

set o

f su

bm

od

els

.

•W

ell-d

efin

ed in

terf

ace

s bet

wee

n th

e su

bmod

els.

•W

ell-d

efin

ed se

ma

ntic

s of t

he c

ombi

ned

mod

el.

•S

ub

mo

de

ls can

be

reus

ed.

111

Col

oure

d P

etri

Net

s 3

Res

ourc

e al

loca

tion

exam

ple

AP

3`(q

,0)

BP

2`(p

,0)

CP

DP

EPT

1[x

=q]

T2

T3

T4

T5

R E

1èD

ecla

ratio

ns:

type

U =

with

p |

q;

type

I =

int;

type

P =

pro

duct

U *

I;ty

pe E

= w

ith e

;va

r x

: U;

var

i : I;

S E

3è

T E

2è

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

if x=

qth

en 1

`(x,

i+1)

else

em

pty

e

if x=

q th

en 1

èel

se e

mpt

y

case

x o

f p

=>

2è

| q =

> 1

è

2è

e

if x=

p th

en 1

èel

se e

mpt

y e

case

x o

f p

=>

2è

| q =

> 1

è

if x=

pth

en 1

`(x,

i+1)

else

em

pty

Col

oure

d P

etri

Net

s 4

Occ

urre

nce

of e

nabl

ed b

indi

ng

SE

33`

e

BP

22`

(p,0

)

T2

CP

SE

33`

e

BP

22`

(p,0

)

T2

CP

Bin

ding

:<

x=p,

i=0>

SE

11`

e

BP

11`

(p,0

)

T2

CP

11`

(p,0

)

New

Mar

king

(x,i)

case

x o

f p

=>

2è

| q =

> 1

è

(x,i)

11`

(p,0

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

22`

e

11`

(p,0

)(x

,i)

(x,i)

case

x o

f p

=>

2è

| q =

> 1

è

(x,i)

112

Col

oure

d P

etri

Net

s 5

Bin

ding

whi

ch is

not

ena

bled

SE

33`

e

BP

22`

(p,0

)

T2

CP

SE

33`

e

BP

22`

(p,0

)

T2

CP

Bin

ding

:<

x=q,

i=2>

(x,i)

case

x o

f p

=>

2è

| q =

> 1

è

(x,i)

11`

(q,2

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

11`

e

11`

(q,2

)(x

,i)

Bin

ding

can

not o

ccur

Col

oure

d P

etri

Net

s 6

A m

ore

com

plex

exa

mpl

e

SE

33`

e

BP

31`

(p,2

)+1`

(p,4

)+1`

(q,3

)

T2

CP

SE

33`

e

BP

31`

(p,2

)+1`

(p,4

)+1`

(q,3

)

T2

CP

Bin

ding

:<

x=p,

i=2>

SE

33`

e

BP

31`

(p,2

)+1`

(p,4

)+1`

(q,3

)

T2

CP

Bin

ding

:<

x=q,

i=3>

(x,i)

case

x o

f p

=>

2è

| q =

> 1

è

(x,i)

11`

(p,2

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

22`

e

11`

(p,2

)(x

,i)

11`

(q,3

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

11`

e

11`

(q,3

)(x

,i)

113

Col

oure

d P

etri

Net

s 7

Con

curr

ency

SE

33`

e

BP

31`

(p,2

)+1`

(p,4

)+1`

(q,3

)

T2

CP

Bin

ding

:<

x=p,

i=2>

SE

BP

11`

(p,4

)

T2

CP

21`

(p,2

)+1`

(q,3

)

Bin

ding

:<

x=q,

i=3>

21`

(p,2

)+1`

(q,3

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

33`

e

21`

(p,2

)+1`

(q,3

)(x

,i)

(x,i)

case

x o

f p

=>

2è

| q =

> 1

è

(x,i)

•T

he tw

o bi

ndin

gs m

ay o

ccur

co

ncu

rre

ntly

.

•T

his

is p

ossi

ble

beca

use

they

use

d

iffe

ren

tto

ken

s.

Col

oure

d P

etri

Net

s 8

Con

flict

SE

33`

e

BP

31`

(p,2

)+1`

(p,4

)+1`

(q,3

)

T2

CP

Bin

ding

:<

x=p,

i=2>

SE

33`

e

BP

31`

(p,2

)+1`

(p,4

)+1`

(q,3

)

T2

CP

Bin

ding

:<

x=p,

i=4>

11`

(p,2

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

22`

e

11`

(p,2

)(x

,i)

11`

(p,4

)(x

,i)

case

x o

f p

=>

2è

| q =

> 1

è

22`

e

11`

(p,4

)(x

,i)

•T

hese

two

bind

ings

can

not o

ccur

co

ncu

rre

ntly

.

•T

he r

easo

n is

that

they

nee

d th

e sa

me

to

ken

s.

114

Col

oure

d P

etri

Net

s 9

Res

ourc

e al

loca

tion

syst

emT

wo

kind

s of

pro

cess

es:

•T

hree

cyc

lic q

-pro

cess

es (

stat

es A

,B,C

,D a

nd E

).

•T

wo

cycl

ic p

-pro

cess

es (

stat

es B

,C,D

and

E).

Thr

ee k

inds

of re

sou

rce

s:

•R

epre

sent

ed b

y th

e pl

aces

R, S

and

T.

Dur

ing

a cy

cle

a pr

oces

s rese

rve

s som

e re

sour

ces

and

rele

ase

s the

m a

gain

:

•T

oken

s ar

e rem

ove

d fro

m a

nd ad

de

d to

the

reso

urce

pla

ces

R, S

and

T.

A c

ycle

co

un

ter is

incr

ease

d ea

ch ti

me

a pr

oces

sco

mpl

etes

a fu

ll cy

cle.

It is

rat

her

stra

ight

forw

ard

to p

rove

that

the

reso

urce

allo

catio

n sy

stem

ca

nn

ot

de

ad

lock

.

•W

hat h

appe

ns if

we

add

an a

dditi

onal

toke

n to

plac

e S

– i.

e., i

f we

star

t with

four

S-r

esou

rces

inst

ead

of th

ree?

Col

oure

d P

etri

Net

s 10

Col

oure

d P

etri

Net

sD

ecl

ara

tion

s:

•T

ype

s, f

un

ctio

ns, o

pe

ratio

ns a

nd v

aria

ble

s.

Eac

h pla

ce h

as th

e fo

llow

ing

insc

riptio

ns:

•N

am

e (fo

r id

entif

icat

ion)

.

•C

olo

ur

set (sp

ecify

ing

the

type

of t

oken

s w

hich

may

res

ide

on th

e pl

ace)

.

•In

itia

l ma

rkin

g (m

ulti-

set o

f tok

en c

olou

rs).

Eac

h tra

nsi

tion

has

the

follo

win

g in

scrip

tions

:

•N

am

e (fo

r id

entif

icat

ion)

.

•G

ua

rd (

bool

ean

expr

essi

on c

onta

inin

g so

me

ofth

e va

riabl

es).

Eac

h arc

has

the

follo

win

g in

scrip

tions

:

•A

rc e

xpre

ssio

n (co

ntai

ning

som

e of

the

varia

bles

).W

hen

the

arc

expr

essi

on is

eva

luat

ed it

yie

lds

am

ulti-

set o

f tok

en c

olou

rs.

115

Col

oure

d P

etri

Net

s 11

Ena

blin

g an

d oc

curr

ence

A b

ind

ing

assi

gns

a co

lou

r (i.e

., a

valu

e) to

eac

hva

ria

ble

of a

tran

sitio

n.

A b

ind

ing

ele

me

nt is

a p

air

(t,b

) w

here

t is

atr

an

sitio

n w

hile

b is

a bi

nd

ing

for

the

varia

bles

of t.

Exa

mpl

e: (

T2,

<x=

p, i=

2>).

A b

indi

ng e

lem

ent i

s en

ab

led i

f and

onl

y if:

•T

here

are

enou

gh to

kens (

of th

e co

rrec

t col

ours

on

each

inpu

t-pl

ace)

.

•T

he g

ua

rd e

valu

ates

to tr

ue.

Whe

n a

bind

ing

elem

ent i

s en

able

d it

may

o

ccu

r:

•A

mul

ti-se

t of t

oken

s is

rem

ove

d fro

m e

ach

inpu

t-pl

ace.

•A

mul

ti-se

t of t

oken

s is

add

ed t

o ea

chou

tput

-pla

ce.

A b

indi

ng e

lem

ent m

ay o

ccur

co

ncu

rre

ntly

to o

ther

bind

ing

elem

ents

– if

f the

re a

re s

o m

any

toke

nsth

at e

ach

bind

ing

elem

ent c

an g

et it

s "o

wn

shar

e".

Col

oure

d P

etri

Net

s 12

Mai

n ch

arac

teris

tics

of C

P-n

ets

Com

bina

tion

of te

xt a

nd g

rap

hic

s.

De

cla

ratio

ns a

nd ne

t in

scrip

tion

s are

spe

cifie

d by

mea

ns o

f a fo

rmal

lang

uage

, e.g

., a

pro

gra

mm

ing

lan

gu

ag

e.

•T

ypes

, fun

ctio

ns, o

pera

tions

, var

iabl

es a

ndex

pres

sion

s.

Ne

t st

ruct

ure

cons

ists

of p

lace

s, tr

ansi

tions

and

arcs

(fo

rmin

g a

bi-p

artit

e gr

aph)

.

•T

o m

ake

a C

P-n

et

rea

da

ble

it is

impo

rtan

t to

mak

e a

nice

gra

phic

al la

yout

.

•T

he g

raph

ical

layo

ut h

as

no

fo

rma

l me

an

ing

.

CP

-net

s ha

ve th

e sa

me

kind

of

con

curr

en

cyp

rop

ert

ies a

s P

lace

/Tra

nsiti

on N

ets.

116

Col

oure

d P

etri

Net

s 13

For

mal

def

initi

on o

f CP

-net

s

Def

initi

on:

A C

olo

ure

d P

etr

i Ne

t is

a t

up

le C

PN

= (

Σ, P

, T

, A

, N

,C

, G, E

, I)

sat

isfy

ing

the

follo

win

g re

quire

men

ts:

(i)Σ

is a

fini

te s

et o

f non

-em

pty

type

s, c

alle

d co

lour

set

s.

(ii)

P is

a fi

nite

set

of pl

aces

.

(iii)

T is

a fi

nite

set

of tr

ansi

tions

.

(iv)

A is

a fi

nite

set

of a

rcs

such

that

:

•P

∩ T

= P

∩ A

= T

∩ A

= Ø

.

(v)

N is

a n

ode

func

tion.

It is

def

ined

from

A in

to P ×

T ∪

T ×

P.

(vi)

C is

a co

lour

func

tion.

It is

def

ined

from

P in

to Σ.

(vii)

G is

a g

uard

fu

nct

ion

. It

is d

efin

ed

fro

m T

into

exp

ress

ion

ssu

ch th

at:

•∀

t ∈T

: [T

ype(

G(t

)) =

Boo

l ∧ T

ype(

Var

(G(t

))) ⊆

Σ].

(viii)

E i

s a

n a

rc e

xpre

ssio

n fu

nct

ion

. It

is

de

fine

d f

rom

A i

nto

expr

essi

ons

such

that

:•

∀a ∈

A: [

Typ

e(E

(a))

= C

(p(a

)) MS ∧

Typ

e(V

ar(E

(a))

) ⊆ Σ

]w

here

p(a

) is

the

plac

e of

N(a

).

(ix)

I is

an

initi

aliz

atio

n

fun

ctio

n.

It i

s d

efin

ed

fro

m P

in

tocl

osed

exp

ress

ions

suc

h th

at:

•∀

p ∈P

: [T

ype(

I(p)

) =

C(p

) MS].

Col

oure

d P

etri

Net

s 14

For

mal

def

initi

on o

f beh

avio

ur

Def

initi

on:

A s

tep

is a

mul

ti-se

t of b

indi

ng e

lem

ents

.

A s

tep

Y is

en

ab

led

in a

ma

rkin

g M

iff

th

e f

ollo

win

g p

rop

ert

y is

satis

fied:

∀p

∈P

:∑

(t,b

) ∈Y E

(p,t)

<b>

≤ M

(p).

Wh

en

a s

tep

Y is

en

ab

led

in a

ma

rkin

g M

1 it

ma

y oc

cur,

cha

ng

ing

the

mar

king

M1

to a

noth

er m

arki

ng M 2

, def

ined

by:

∀p

∈P

: M

2(p)

= (

M1(

p) –

∑(t

,b) ∈

Y E

(p,t)

<b>

) +

∑(t

,b) ∈

Y E

(t,p

)<b>

.

Th

e f

irst

su

m i

s ca

lled

th

e

rem

oved

to

ken

s w

hile

th

e s

eco

nd

is

calle

d t

he

ad

de

d to

ken

s. M

ore

ove

r w

e s

ay

tha

t M

2 is

dir

ect

lyre

acha

ble

from

M1

by t

he o

ccur

renc

e of

the

ste

p Y

, w

hich

we

also

deno

te: M

1 [Y› M

2.

An

occu

rren

ce s

eque

nce is

a s

eque

nce

of m

arki

ngs

and

step

s:

M1 [Y

1› M

2 [Y

2› M

3 …

Mn [Y

n› M

n+1

such

th

at

Mi [

Yi› M

i+1

fo

r a

ll i ∈

1. . n

. W

e t

he

n s

ay

tha

t M n+

1 is

reac

habl

e fr

om

M1.

We

use

[M ›

to

de

no

te t

he

se

t o

f m

ark

ing

sw

hich

are

rea

chab

le fr

om M

.

117

Col

oure

d P

etri

Net

s 15

For

mal

def

initi

onT

he e

xist

ence

of a

fo

rma

l de

finiti

on i

s ve

ryim

port

ant:

•It

is th

e ba

sis

for si

mu

latio

n, i.e

., ex

ecut

ion

of th

eC

P-n

et.

•It

is a

lso

the

basi

s fo

r th

e form

al v

erific

atio

nm

etho

ds (

e.g.

, sta

te s

pace

s an

d pl

ace

inva

riant

s).

•W

ithou

t the

form

al d

efin

ition

, it w

ould

hav

ebe

en im

poss

ible

to o

btai

n a so

un

d net

cla

ss.

It is

no

t n

ece

ssa

ry for

a u

ser t

o kn

ow th

e fo

rmal

defin

ition

of C

P-n

ets:

•T

he c

orre

ct sy

nta

x is

chec

ked

by th

e C

PN

edi

tor,

i.e.,

the

com

pute

r to

ol b

y w

hich

CP

-net

s ar

eco

nstr

ucte

d.

•T

he c

orre

ct u

se o

f the

se

ma

ntic

s (i.e

., th

een

ablin

g ru

le a

nd th

e oc

curr

ence

rul

e) is

guar

ante

ed b

y th

e C

PN

sim

ulat

or a

nd th

eC

PN

tool

s fo

r fo

rmal

ver

ifica

tion.

Col

oure

d P

etri

Net

s 16

Hig

h-le

vel c

ontr

a lo

w-le

vel n

ets

The

rel

atio

nshi

p be

twee

n C

P-n

ets

and

Pla

ce/T

rans

ition

Net

s (P

T-n

ets)

is

an

alo

go

us t

oth

e re

latio

nshi

p be

twee

n hi

gh-le

vel p

rogr

amm

ing

lang

uage

s an

d as

sem

bly

code

.

•In

theo

ry, t

he tw

o le

vels

hav

e ex

actly

the

sam

eco

mp

uta

tion

al p

ow

er.

•In

pra

ctic

e, h

igh-

leve

l lan

guag

es h

ave

muc

hm

ore

mo

de

llin

g p

ow

er –

bec

ause

they

hav

ebe

tter

stru

ctur

ing

faci

litie

s, e

.g.,

type

s an

dm

odul

es.

Eac

h C

P-n

et h

as a

n equ

iva

len

t PT

-net

– a

nd v

ice

vers

a.

•T

his

equi

vale

nce

is u

sed

to d

eriv

e th

e de

finiti

onof

ba

sic

pro

pe

rtie

s and

to e

stab

lish

the

verific

atio

n m

eth

od

s.

•In

pra

ctic

e, w

e ne

ver t

rans

late

a C

P-n

et in

to a

PT

-net

– o

r vi

ce v

ersa

.

•D

escr

iptio

n, s

imul

atio

n an

d ve

rific

atio

n ar

e do

ned

ire

ctly

in te

rms

of C

P-n

ets.

118

Col

oure

d P

etri

Net

s 17

Oth

er k

inds

of h

igh-

leve

l net

sC

P-n

ets

have

bee

n de

velo

ped

from

Pre

dica

te/T

rans

ition

Net

s.

•H

art

ma

nn

Ge

nrich

& K

urt

La

ute

nb

ach.

•W

ith r

espe

ct to

desc

rip

tion

and

sim

ula

tion t

hetw

o m

odel

s ar

e ne

arly

iden

tical

.

•W

ith r

espe

ct to

form

al v

erific

atio

n the

re a

reso

me

diffe

renc

es.

Sev

eral

oth

er k

inds

of h

igh-

leve

l Pet

ri N

ets

exis

t.

•T

hey

all b

uild

upo

n th

e sa

me ba

sic

ide

as, b

ut u

sed

iffe

ren

t la

ng

ua

ge

s for

decl

arat

ions

and

insc

riptio

ns.

•A

det

aile

d co

mpa

rison

is o

utsi

de th

e sc

ope

ofth

is ta

lk.

Col

oure

d P

etri

Net

s 18

Sim

ple

prot

ocol

Sen

dP

acke

tT

rans

mit

Pac

ket

Rec

eive

Pac

ket

Rec

eive

Ack

now

.T

rans

mit

Ack

now

.

Sen

d

INT

xDA

TA

1`(1

,"M

odel

lin")

+1`

(2,"

g an

d A

n")+

1`(3

,"al

ysis

b")

+1`

(4,"

y M

eans

")+

1`(5

,"of

Col

ou")

+1`

(6,"

red

Pet

r")+

1`(7

,"i N

ets#

#")+

1`(8

,"##

####

##")

Nex

tSen

dIN

T

1

DIN

TA

INT

xDA

TA

Rec

eive

dD

AT

A

""

Nex

tRec

INT

1

B

INT

xDA

TA

CIN

T

RP

8

Int_

0_10

RA

Int_

0_108

Sen

der

Net

wo

rkR

ecei

ver

type

INT

= in

t;ty

pe B

OO

L =

boo

l;ty

pe D

AT

A =

str

ing;

ty

pe IN

TxD

AT

A =

pro

duct

INT

* D

AT

A;

var

n, k

: IN

T;

var

p,st

r : D

AT

A;

val s

top

= "

####

####

";

type

Int_

0_10

= in

t with

0..1

0;ty

pe In

t_1_

10 =

int w

ith 1

..10;

var

s : I

nt_0

_10;

var

r : I

nt_1

_10;

fun

Ok(

s : I

nt_0

_10,

r :

Int_

1_10

) =

(r≤

s);

(n,p

)(n

,p)

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

(n,p

)

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

kn

k

if n=

kth

en k

+1

else

k

if O

k(s,

r)th

en 1

`nel

se e

mpt

y

n

kn

str

ssn

119

Col

oure

d P

etri

Net

s 19

Sen

d pa

cket

Sen

d

INT

xDA

TA

8

1`(1

,"M

odel

lin")

+ 1

`(2,

"g a

nd A

n")

+ 1

`(3,

"aly

sis

b")

+ 1

`(4,

"y M

eans

")

+ 1

`(5,

"of C

olou

")+

1`(

6,"r

ed P

etr"

)+

1`(

7,"i

Net

s##"

)+

1`(

8,"#

####

###"

)

Sen

dP

acke

tA

INT

xDA

TA

Nex

tSen

dIN

T

1

11`

1(n,p

)

(n,p

) n

Onl

y th

e bi

ndin

g <

n=1,

p= "M

odel

lin">

is e

na

ble

d.

•W

hen

the

bind

ing o

ccu

rs it

add

s a

toke

n to

plac

e A

. The

toke

n re

pres

ents

that

the

pack

et(1

,"M

odel

lin")

is s

ent t

o th

e ne

twor

k.

•T

he p

acke

t is n

ot r

emov

ed fr

om p

lace

S

en

d and

the

Ne

xtS

en

d cou

nter

is no

t cha

nged

.

Col

oure

d P

etri

Net

s 20

Tra

nsm

it pa

cket

A

INT

xDA

TA

11`

(1,"

Mod

ellin

")

Tra

nsm

itP

acke

t

RP

8

Int_

0_10

11`

8

B

INT

xDA

TA

(n,p

)

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

s

The

re a

re n

ow 10

en

ab

led

bin

din

gs:

•T

hey

are

all o

f the

form

<n=

1, p

= "M

odel

lin",

s=

8, r

=…

>.

•T

he v

aria

ble

r ca

n ta

ke

10

diff

ere

nt

valu

es,

beca

use

the

type

of r

is d

efin

ed to

con

tain

the

inte

gers

1. .1

0.

The

fun

ctio

n O

k(s,

r) c

heck

s w

heth

er r

≤

s.

•F

or r

∈1.

.8, O

k(s,

r) =

tru

e. T

his

mea

ns th

at th

eto

ken

is m

oved

from

A to

B, i

.e.,

that

the

pack

et is

succ

ess

fully

tra

nsm

itte

d ove

r th

e ne

twor

k.

•F

or r

∈9.

.10,

Ok(

s,r)

= f

als

e. T

his

mea

ns th

at n

oto

ken

is a

dded

to B

, i.e

., th

at th

e pa

cket

is

lost

.

•T

he C

PN

sim

ulat

or m

ake ra

nd

om

cho

ices

betw

een

enab

led

bind

ings

. Hen

ce th

ere

is

80

%ch

ance

for s

ucc

ess

ful t

ran

sfe

r.

120

Col

oure

d P

etri

Net

s 21

Rec

eive

pac

ket

B

INT

xDA

TA

11`

(1,"

Mod

ellin

")

Rec

eive

Pac

ket

Nex

tRec

INT1

11`

1

Rec

eive

dD

AT

A

""1

1`""

CIN

T

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

k

if n=

kth

en k

+1

else

k

str

It is

che

cked

whe

ther

the

num

ber

of th

e in

com

ing

pack

et n

ma

tch

es t

he n

umbe

r of

the

expe

cted

pack

et k

.

Col

oure

d P

etri

Net

s 22

Cor

rect

pac

ket n

umbe

r

B

11`

(3,"

alys

is b

")

Rec

eive

Pac

ket

Nex

tRec

11`

3

Rec

eive

d1

1`"M

odel

ling

and

An"

C

(n,p

)1

1`(3

,"al

ysis

b")

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

11`

"Mod

ellin

g an

d A

naly

sis

b"

if n=

kth

en k

+1

else

k

11`

4

k

11`

3

if n=

kth

en k

+1

else

k

11`

4

str1

1`"M

odel

ling

and

An"

•T

he d

ata

in th

e pa

cket

is

con

cate

na

ted to

the

data

alre

ady

rece

ived

.

•T

he N

ext

Re

c cou

nter

is in

crea

sed

by o

ne.

•A

n a

ckn

ow

led

ge

me

nt

me

ssa

ge

is s

ent.

It co

ntai

nsth

e nu

mbe

r of

the

next

pac

ket w

hich

the

rece

iver

wan

ts to

get

.

121

Col

oure

d P

etri

Net

s 23

Wro

ng p

acke

t num

ber

B

11`

(2,"

g an

d A

n") R

ecei

veP

acke

tN

extR

ec1

11`

3

Rec

eive

d1

1`"M

odel

ling

and

An"

C

(n,p

)1

1`(2

,"g

and

An"

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

11`

"Mod

ellin

g an

d A

n"

if n=

kth

en k

+1

else

k

11`

3

k

11`

3

if n=

kth

en k

+1

else

k

11`

3

str1

1`"M

odel

ling

and

An"

•T

he d

ata

in th

e pa

cket

is

ign

ore

d.

•T

he N

ext

Re

c cou

nter

is u

ncha

nged

.

•A

n a

ckn

ow

led

ge

me

nt

me

ssa

ge

is s

ent.

It co

ntai

nsth

e nu

mbe

r of

the

next

pac

ket w

hich

the

rece

iver

wan

ts to

get

.

Col

oure

d P

etri

Net

s 24

Tra

nsm

it ac

know

ledg

emen

t

CIN

T

11`

2

Tra

nsm

itA

ckno

w.

DIN

T

RA

Int_

0_10

81

1`8

nif

Ok(

s,r)

then

1`n

else

em

pty

s

Thi

s tr

ansi

tion

wor

ks in

a s

imila

r w

ay a

sT

ran

smit

Pa

cke

t.

•T

he to

ken

on p

lace

R

A de

term

ines

the

succ

ess

rate

for

tran

smis

sion

of a

ckno

wle

dgem

ents

.

•W

hen

RA

cont

ains

a to

ken

with

val

ue 8

, the

succ

ess

rate

is

80

%.

•W

hen

RA

cont

ains

a to

ken

with

val

ue 1

0,

no

ackn

owle

dgem

ents

are

lost

.

•W

hen

RA

cont

ains

a to

ken

with

val

ue 0

, a

llac

know

ledg

emen

ts a

re lo

st.

122

Col

oure

d P

etri

Net

s 25

Rec

eive

ack

now

ledg

emen

t

Rec

eive

Ack

now

.

Nex

tSen

dIN

T

1

11`

1

DIN

T

11`

2

n

kn

Whe

n an

ack

now

ledg

emen

t a

rriv

es

to th

e Se

nd

er it

is u

sed

to u

pdat

e th

e Ne

xtS

en

d cou

nter

.

•In

this

cas

e th

e co

unte

r va

lue

beco

mes

2, a

ndhe

nce

the S

en

de

r will

beg

in to

sen

d pa

cket

num

ber

2.

Col

oure

d P

etri

Net

s 26

Inte

rmed

iate

sta

te

Sen

dP

acke

tT

rans

mit

Pac

ket

Rec

eive

Pac

ket

Rec

eive

Ack

now

.T

rans

mit

Ack

now

.

Sen

d

INT

xDA

TA

8

1`(1

,"M

odel

lin")

+ 1

`(2,

"g a

nd A

n")

+ 1

`(3,

"aly

sis

b")

+ 1

`(4,

"y M

eans

")

+ 1

`(5,

"of C

olou

")+

1`(

6,"r

ed P

etr"

)+

1`(

7,"i

Net

s##"

)+

1`(

8,"#

####

###"

)

Nex

tSen

dIN

T

1

11`

5

DIN

T

11`

6

A

INT

xDA

TA

Rec

eive

dD

AT

A

""1

1`"M

odel

ling

and

Ana

lysi

s by

Mea

ns

of C

olou

"

Nex

tRec

INT

1

11`

6

B

INT

xDA

TA

11`

(5,"

of C

olou

")

CIN

T

11`

6

Sen

der

Net

wo

rkR

ecei

ver

RP

8

Int_

0_10

11`

8

RA

Int_

0_10

81

1`8

(n,p

)(n

,p)

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

(n,p

)

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

n

k

if n=

kth

en k

+1

else

k

if O

k(s,

r)th

en 1

`nel

se e

mpt

y

n

kn

n

str

ss

•T

he R

ece

ive

r is e

xpec

ting

pack

age

no. 6

. Thi

sm

eans

that

it h

as s

ucce

ssfu

lly r

ecei

ved

the

first

5pa

cket

s.

•T

he S

en

de

r is s

till s

endi

ng p

acke

t no.

5. I

n a

mom

ent i

t will

rec

eive

an

ackn

owle

dgem

ent

cont

aini

ng a

req

uest

for

pack

et n

o. 6

.

•W

hen

the

ackn

owle

dgem

ent i

s re

ceiv

ed th

eN

ext

Se

nd c

ount

er is

upd

ated

and

the

Se

nd

er w

illst

art s

endi

ng p

acke

t no.

6.

123

Col

oure

d P

etri

Net

s 27

Fin

al s

tate

Sen

dP

acke

tT

rans

mit

Pac

ket

Rec

eive

Pac

ket

Rec

eive

Ack

now

.T

rans

mit

Ack

now

.

Sen

d

INT

xDA

TA

8

1`(1

,"M

odel

lin")

+ 1

`(2,

"g a

nd A

n")

+ 1

`(3,

"aly

sis

b")

+ 1

`(4,

"y M

eans

")

+ 1

`(5,

"of C

olou

")+

1`(

6,"r

ed P

etr"

)+

1`(

7,"i

Net

s##"

)+

1`(

8,"#

####

###"

)

Nex

tSen

dIN

T

1

11`

9

DIN

TA

INT

xDA

TA

Rec

eive

dD

AT

A

""1

1`"M

odel

ling

and

Ana

lysi

s by

Mea

ns

of C

olou

red

Pet

ri N

ets#

#" Nex

tRec

INT

1

11`

9

B

INT

xDA

TA

CIN

T

Sen

der

Net

wo

rkR

ecei

ver

RP

8

Int_

0_10

11`

8

RA

Int_

0_10

81

1`8

(n,p

)(n

,p)

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

(n,p

)

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

n

k

if n=

kth

en k

+1

else

k

if O

k(s,

r)th

en 1

`nel

se e

mpt

y

n

kn

n

str

ss

•W

hen

the

last

pac

ket,

i.e.,

pack

et n

o. 8

rea

ches

the

Re

ceiv

er a

n ac

know

ledg

emen

t with

val

ue 9

isse

nt.

•W

hen

this

ack

now

ledg

emen

t rea

ches

the

Se

nd

er

the

Ne

xtS

en

d cou

nter

is u

pdat

ed to

9.

•T

his

mea

ns th

at th

e Se

nd

Pa

cke

t tran

sitio

n no

long

er c

an o

ccur

, and

hen

ce th

e tr

ansm

issi

onst

ops.

Col

oure

d P

etri

Net

s 28

Par

t 2: H

iera

rchi

cal C

P-n

ets

A h

iera

rchi

cal C

P-n

et c

onta

ins

a nu

mbe

r of

inte

rre

late

d s

ub

ne

ts– ca

lled p

ag

es.

D

INT

xIN

T

A

INT

xDA

TA

Rec

eive

dD

AT

A

""

B

INT

xDA

TA

CIN

T

Sen

der

HS

Sen

der

Net

wor

k

HS

Net

wor

k

Rec

eive

r

HS

Rec

eive

r

Sim

ple

Pro

toco

l

Sen

dP

acke

t

Nex

tSen

dIN

T

1

Rec

eive

Ack

now

.

Sen

d

INT

xDA

TA

DIN

TA

INT

xDA

TA

Tra

nsm

itP

acke

t

RP

8

Int_

0_10

RA

Int_

0_10

8 Tra

nsm

itA

ckno

w.

DIN

T

CIN

T

A

INT

xDA

TA

B

INT

xDA

TA

Nex

tRec

INT

1R

ecei

veP

acke

t

CIN

T

Rec

eive

dD

AT

A

""

B

INT

xDA

TA

Sen

der

Rec

eive

r N

etw

ork

Out

Out

Out

In

In

InO

ut

In

I/O

(n,p

)

kn

n

(n,p

)

n

s

s

(n,p

)

n

if O

k(s,

r)th

en 1

`nel

se e

mpt

y

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

k if n=

kth

en k

+1

else

k

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

str

124

Col

oure

d P

etri

Net

s 29

Sub

stitu

tion

tran

sitio

nsA

pag

e m

ay c

onta

in o

ne o

re m

ore

sub

stitu

tion

tra

nsi

tion

s.

•E

ach

subs

titut

ion

tran

sitio

n is

rel

ated

to a

p

ag

e,i.e

., a s

ub

ne

t pro

vidi

ng a

mo

re d

eta

iled

de

scrip

tion t

han

the

tran

sitio

n its

elf.

•T

he p

age

is a

sub

pa

ge o

f the

sub

stitu

tion

tran

sitio

n.

The

re is

a we

ll-d

efin

ed

inte

rfa

ce be

twee

n a

subs

titut

ion

tran

sitio

n an

d its

sub

page

:

•T

he p

lace

s su

rrou

ndin

g th

e su

bstit

utio

n tr

ansi

tion

are

sock

et

pla

ces.

•T

he s

ubpa

ge c

onta

ins

a nu

mbe

r of

p

ort

pla

ces.

•S

ocke

t pla

ces

are re

late

d to

por

t pla

ces

– in

asi

mila

r w

ay a

s ac

tual

par

amet

ers

are

rela

ted

tofo

rmal

par

amet

ers

in a

pro

cedu

re c

all.

•A

soc

ket p

lace

has

alw

ays

the

sam

e m

ark

ing a

sth

e re

late

d po

rt p

lace

. The

two

plac

es a

re ju

std

iffe

ren

t vi

ew

s of t

he s

ame

plac

e.

Su

bst

itutio

n t

ran

sitio

ns w

ork

in a

sim

ilar

way

as

the

refin

emen

t prim

itive

s fo

und

in m

any

syst

emde

scrip

tion

lang

uage

s –

e.g.

, SA

DT

dia

gram

s.

Col

oure

d P

etri

Net

s 30

Pag

es c

an b

e us

ed m

ore

than

onc

e

D

INT

xIN

T

A

INT

xDA

TA

Rec

eive

dD

AT

A

""

B1

INT

xDA

TA

C1

INT

Sen

der

HS

Sen

der

Net

wor

k

HS

Net

wor

k

Rec

No1

HS

Rec

eive

rB

1->

BC

1->

C

B2

INT

xDA

TA

C2

INT

Rec

No2

HS Rec

eive

rB

2->

BC

2->

C

Rec

eive

dD

AT

A

""

Sim

ple

Pro

toco

l wit

h 2

Rec

eive

rs

Sen

dP

acke

t

Nex

tSen

dIN

T

1

Rec

eive

Ack

now

.

Sen

d

INT

xDA

TA

DIN

TxI

NT

A

INT

xDA

TA

Tra

nsm

itP

acke

t

RP

8

Int_

0_10

RA

1In

t_0_

10

8 Tra

nsm

itA

ckno

w.

DIN

TxI

NT

C1

INTB2

INT

xDA

TA

A

INT

xDA

TA

B1

INT

xDA

TA

RA

2In

t_0_

108Tra

nsm

itA

ckno

w.

C2

INT

Nex

tRec

INT

1R

ecei

veP

acke

t

CIN

T

Rec

eive

dD

AT

A

""

B

INT

xDA

TA

Sen

der

Rec

eive

r N

etw

ork

Out

Out

Out

Out

In

In

In

Out

In

In

I/O

(n,p

)

km

in(n

1,n2

)

n

(n,p

)

1`(n

1,1)

+1`

(n2,

2)

ss

(n,p

)

if O

k(s,

r2)

then

1`(

n,p)

else

em

pty n

if O

k(s,

r)th

en 1

`(n,

1)el

se e

mpt

y

if O

k(s,

r1)

then

1`(

n,p)

else

em

pty

s

nif

Ok(

s,r)

then

1`(

n,2)

else

em

pty

k if n=

kth

en k

+1

else

k

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

str

The

re a

re tw

o di

ffere

nt in

sta

nce

s of t

he R

ece

ive

rpa

ge. E

ach

inst

ance

has

its

ow

n m

ark

ing

.

125

Col

oure

d P

etri

Net

s 31

Rin

g ne

twor

k

4to1

PA

CK

1to2

PA

CK

Site

1

HS

Site

4to1

->In

com

ing

1to2

->O

utgo

ing

2to3

PA

CK

Site

2

HS

Site

1to2

->In

com

ing

2to3

->O

utgo

ing

3to4

PA

CK

Site

3H

S

Site

2to3

->In

com

ing

3to4

->O

utgo

ing

Site

4H

S

Site

3to4

->In

com

ing

4to1

->O

utgo

ing

Rin

g N

etw

ork

New

Pac

k

Pac

kNo

INT

1

Pac

kage

PA

CK

Sen

d

Out

goin

gPA

CK O

ut

Inco

min

gP

AC

KIn

Rec

eive

Sit

e

Rec

eive

d

PA

CK

n+1

n

{se=

this

(), r

e=r,

no=

n}

p

if #r

e p

<>

this

()th

en 1

`pel

se e

mpt

y

p

if #r

e p

<>

this

()th

en 1

`pel

se e

mpt

y

if #r

e p

= th

is()

then

1`p

else

em

pty

if #r

e p

= th

is()

then

1`p

else

em

pty

Col

oure

d P

etri

Net

s 32

For

mal

def

initi

on o

f hie

rarc

hica

lC

P-n

ets

The

syn

tax a

nd se

ma

ntic

s of h

iera

rchi

cal C

P-n

ets

have

form

al d

efin

itio

ns –

sim

ilar

to th

e de

finiti

ons

for

non-

hier

arch

ical

CP

-net

s

Eac

h hi

erar

chic

al C

P-n

et h

as a

n e

qu

iva

len

tno

n-hi

erar

chic

al C

P-n

et –

and

vic

e ve

rsa.

•T

he tw

o ki

nds

of n

ets

have

the

sam

eco

mp

uta

tion

al p

ow

er – b

ut h

iera

rchi

cal C

P-n

ets

have

muc

h m

ore m

od

elli

ng

po

we

r.

•T

he e

qu

iva

len

ce is

use

d fo

r the

ore

tica

l pu

rpo

ses.

•In

pra

ctic

e, w

e ne

ver t

rans

late

a h

iera

rchi

cal

CP

-net

into

a n

on-h

iera

rchi

cal C

P-n

et –

or

vice

vers

a.

126

Col

oure

d P

etri

Net

s 33

CP

-net

s m

ay b

e la

rge

A ty

pica

l in

du

stria

l ap

plic

atio

n of C

P-n

ets

cont

ains

:

•10

-200

pag

es.

•50

-100

0 pla

ces

an

d t

ran

sitio

ns.

•10

-200

colo

ur

sets

.

Thi

s co

rres

pond

s to

th

ou

san

ds/

mill

ion

s o

f n

od

es

ina

Pla

ce/T

rans

ition

Net

.

Mos

t of t

he in

dust

rial a

pplic

atio

ns w

ould

be

tota

llyim

po

ssib

le w

ithou

t:

•C

olou

rs.

•H

iera

rchi

es.

•C

ompu

ter

tool

s.

Col

oure

d P

etri

Net

s 34

Pro

toco

l for

tele

phon

e ne

twor

kT

rans

port

laye

r of

a p

roto

col f

or

dig

ital t

ele

ph

on

eco

mm

un

ica

tion.

ISD

N#

1

US

ER

_T

OP

#2

NU

LL

#3

DE

CL

AR

E#

4

CA

LL

_R

EC

#1

1

CO

NN

EC

T#

12

INC

OM

ING

#1

3

HIE

RA

RC

HY

#1

00

10

CA

LL

_IN

I#6

OV

ER

LA

P#

9

OU

TG

OIN

G#15

CA

LL

_D

EL

#1

6

NU

LL

_S

ET

#5

AC

TIV

E#

7

DIS

CO

NN

E#

8

RE

LE

AS

E#

17

DIS

C_

IND

#1

8

NE

T_

TO

P#

19

NU

LL

#2

0U

_S

ET

UP

#2

1

N_

SE

TU

P#

22

RO

UT

ING

#2

4

OU

TG

OIN

G#26 N

_E

_P

AR

T#

27

CA

LL

_D

EL

#2

8

OV

ER

LA

P#

29

CO

NN

EC

T#

30

N_

D_

PA

RT

#3

1

CA

LL

_R

EC

#3

2

DIS

CO

NN

E#

33

DIS

CO

NN

E#

34

RE

LE

AS

E#

35

AC

TIV

E#

36

INC

OM

ING

#3

7

CA

LL

_P

RE

#3

8

UR

EQ

_G

EN

#3

9

U_

DIS

C#

23

U_

RE

L#

25

U_

RE

L_

CO

#4

0

U_

PR

OG

#4

1

U_

INF

O#

42

N_

HO

LD

#4

4

U_

HO

LD

#4

5

Fig

. 7

.1

{

Use

rs

U1

U7

U8

U9

U0

U2

U3

U4

U1

0

U1

1

U1

9

U1

2

N0

N3

N4

N2

N8

N7

N1

1

N1

2

N1

9

N1

0

N9

N6

Ne

two

rks

Ove

rvie

w o

f the

hie

rarc

hy s

truc

ture

:

•E

ach n

od

e re

pres

ents

a

pa

ge,

i.e.

, a s

ubne

t.

•E

ach a

rc r

epre

sent

s a tr

an

sitio

n s

ub

stitu

tion

.

127

Col

oure

d P

etri

Net

s 35

Tw

o of

the

mos

t abs

trac

t pag

es

Use

rToN

etw

orkM

essa

ge

Net

wor

kToU

ser Mes

sage

Use

rsN

etw

orks

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

Net

wor

kToU

ser

Mes

sage

Use

rToN

etw

ork

Mes

sage

IntU

serR

eqUIn

tReq

U1

U2

U3

U4

U7

U8

U9

U10

U11

U12

U19

U0

UR

EQ

In

Out

Col

oure

d P

etri

Net

s 36

Typ

ical

pag

e fo

r th

e us

er s

ite

(u,{

mt=

DIS

C,

cr=

cref

,ai

=nu

ll})

(u,{

mt =

RE

L_C

OM

,cr

= c

ref,

ai =

nul

l})

(u, {

mt=

ST

AT

US

,cr

=cr

ef,

ai=

Sta

tus

8})(u

,m)

(u,m

)

(u,c

ref,b

,s)

(u,c

ref,b

,s)

(u,c

ref,b

,s)

(u,0

,non

e,s)

(u,{

mt=

CLE

AR

_RE

Q,

ai=

Cal

lref c

ref}

)

U8

Use

rSta

te

Net

wor

kToU

ser

Mes

sage

In

[#m

t m =

ST

AT

US

_EN

Q,

cref

= #

cr m

]

[#m

t m =

RE

L_C

OM

,cr

ef=

#cr

m]

[#m

t m =

DIS

C,

cref

= #

cr m

]

[#m

t m =

CO

NN

_AC

K,

cref

= #

cr m

]

Use

rToN

etw

ork

Mes

sage

Out

U11

U10

U12

U0

[#m

t m =

RE

L,cr

ef=

#cr

m]

CLE

AR

_RE

Q

Inte

rnal

Use

rReq U

IntR

eq

In(u

,cre

f,b,s

)

Thi

s pa

ge d

escr

ibes

the po

ssib

le a

ctio

ns th

at c

anha

ppen

whe

n th

e us

er s

ite is

in s

tate

U

8:

•F

rom

the n

etw

ork fi

ve d

iffer

ent k

inds

of

mes

sage

s m

ay b

e re

ceiv

ed.

•In

add

ition

ther

e is

one

kin

d of

in

tern

al u

ser

req

ue

st.

•In

thre

e of

the

case

s a ne

w m

ess

ag

e is s

ent t

o th

en

etw

ork

site

.

128

Col

oure

d P

etri

Net

s 37

Typ

ical

tran

sitio

n

(u, {

mt=

ST

AT

US

,cr

=cr

ef,

ai=

Sta

tus

8})

(u,m

)

(u,c

ref,b

,s)

Net

wor

kToU

ser

Mes

sage

[#m

t m =

ST

AT

US

_EN

Q,

cref

= #

cr m

]

Use

rToN

etw

ork M

essa

ge

U8

Use

rSta

te

type

Use

rSta

te =

prod

uct U

ser

* C

allR

ef

*

BC

hanN

ame

* H

oldS

tatu

s;

type

Mes

sage

Rec

=

r

ecor

d m

t : M

essa

geT

ype

* c

r : C

allR

efer

ence

* a

i : M

essa

geD

ata;

type

Mes

sage

=

p

rodu

ct U

ser

* M

essa

geR

ec;

Thi

s tr

ansi

tion

desc

ribes

the ac

tion

s tha

t are

take

nw

hen

a Sta

tus

En

qu

iry m

essa

ge is

rec

eive

d in

sta

teU

8:

•T

he g

uard

che

cks

that

the

mes

sage

is a

S

tatu

sE

nq

uiry

mes

sage

. It a

lso

chec

ks th

at th

e C

all

Re

fere

nce

is c

orre

ct (

i.e.,

mat

ches

the

one

in th

eU

ser

Sta

te to

ken

at p

lace

U8)

.

•A

Sta

tus

me

ssa

ge is s

ent t

o th

e ne

two

rk s

ite. It

tells

that

the

user

site

is in

sta

te U

8.

Col

oure

d P

etri

Net

s 38

SD

L de

scrip

tion

of u

ser

page

CO

NN

AC

KD

ISC

RE

LR

EL

CO

MS

TA

TU

SE

NQ

AN

YO

TH

ER

ME

SS

AG

E

CLE

AR

RE

QU

ES

T

8 C

ON

NE

CT

RE

QU

ES

T

DIS

C

11

CO

NN

EC

TB

-CH

AN

NE

LT

O U

SE

R

10

12R

ELE

AS

EC

RV

RE

LEA

SE

CR

V

RE

LC

OM 0

0

FO

RM

CA

US

EC

ALL

ST

AT

E(U

8 O

R U

10)

ST

AT

US

8

OP

TIO

N

Eac

h ve

rtic

al s

trin

g o

f S

DL

sym

bo

ls des

crib

es a

sequ

ence

of a

ctio

ns –

whi

ch is

tran

slat

ed in

to a

sin

gle

CP

N t

ran

sitio

n.

•T

he tr

an

sla

tion

from

SD

L to

CP

N w

as d

one

ma

nu

ally

.

•T

he tr

ansl

atio

n is

str

aigh

tforw

ard

and

it co

uld

easi

ly b

e au

tom

ate

d.

The

gra

phic

al s

hape

of a

nod

e ha

s a

we

ll-d

efin

ed

mea

ning

in S

DL.

•In

the

CP

-net

the

shap

e is

ret

aine

d –

to im

prov

eth

e re

ad

ab

ility

. It h

as n

o fo

rmal

mea

ning

.

129

Col

oure

d P

etri

Net

s 39

Typ

ical

pag

e fo

r th

e ne

twor

k si

te

N6

Net

Sta

te

Net

Fro

mN

et

NIn

tReq

Use

rToN

etw

ork

Mes

sage

"E"

[#m

t m

= A

LE

RT

, c

ref=

#cr

m]

Net

ToN

et NIn

tReq N

etw

orkT

oUse

r

Mes

sage

"D"

FIR

ST

TIM

EO

UT

T

303

T30

3

NT

imer

Cou

nt

NB

CB

Cha

n

N0

Net

Sta

teN

7N

etS

tate

[#m

t m

=

CA

LL

_PR

OC

,cr

ef=

#cr

m]

[#m

t m

=

RE

L_C

OM

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Sim

ilar

stru

ctur

e as

for

the

user

pag

e –

but s

light

lym

ore

com

plex

.

Col

oure

d P

etri

Net

s 40

SD

L de

scrip

tion

of n

etw

ork

page

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LER

TR

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LLIN

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Sim

ilar

stru

ctur

e as

for

the

user

pag

e –

but s

light

lym

ore

com

plex

.

It is

eas

y to

see

that

ther

e is

a v

ery

stra

ight

forw

ard

rela

tions

hip

betw

een

the SD

L p

ag

e and

the

corr

espo

ndin

g CP

N p

ag

e.

130

Col

oure

d P

etri

Net

s 41

Som

e pa

ges

are

used

man

y tim

esIS

DN

#1

US

ER

_T

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LL

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DE

CL

AR

E#

4

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LL

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1

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NN

EC

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OM

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RA

RC

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00

10

CA

LL

_IN

I#6

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ER

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9

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TG

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6

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7

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8

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LE

AS

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#1

8

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T_

TO

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19

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LL

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SE

TU

P#

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4

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Fig

. 7

.1

{

Use

rs

U1

U7

U8

U9

U0

U2

U3

U4

U1

0

U1

1

U1

9

U1

2

N0

N3

N4

N2

N8

N7

N1

1

N1

2

N1

9

N1

0

N9

N6

Ne

two

rks

•43

pag

es w

ith m

ore

than

100

pag

e in

stan

ces.

•T

he e

ntire

mod

ellin

g of

this

– fa

irly

com

plex

prot

ocol

– w

as m

ade

in o

nly 3

we

eks

(by

a si

ngle

pers

on).

•A

ccor

ding

to e

ngin

eers

at t

he p

artic

ipat

ing

tele

com

mun

icat

ions

com

pany

, the

CP

N m

odel

was

the m

ost

de

taile

d beh

avio

ural

mod

el th

atth

ey h

ad e

ver

seen

for

such

pro

toco

ls.

Col

oure

d P

etri

Net

s 42

Pra

ctic

al u

se o

f CP

-net

sC

P-n

ets

are

used

in

ma

ny

diff

ere

nt

are

as. A

few

sele

cted

exa

mpl

es a

re:

•C

omm

unic

atio

n pr

otoc

ols

(B

RI,

DQ

DB

, AT

M).

•V

LSI c

hips

(cl

ocke

d an

d se

lf-tim

ed).

•B

anki

ng p

roce

dure

s (c

heck

pro

cess

ing

and

fund

str

ansf

er).

•C

orre

ctne

ss o

f AD

A p

rogr

ams

(ren

dezv

ous

stru

ctur

e).

•T

eles

hopp

ing

syst

ems.

•M

ilita

ry s

yste

ms

(rad

ar c

ontr

ol p

ost a

nd n

aval

vess

el).

•S

ecur

ity s

yste

ms

(intr

usio

n al

arm

s, e

tc.)

.

•F

lexi

ble

man

ufac

turin

g.

131

Col

oure

d P

etri

Net

s 43

Sum

mar

y of

pra

ctic

al e

xper

ienc

esG

rap

hic

al r

ep

rese

nta

tion a

nd e

xecu

tab

ility

are

extr

emel

y im

port

ant.

Mos

t pra

ctic

al m

odel

s ar

e larg

e.

•T

hey

cann

ot b

e co

nstr

ucte

d w

ithou

t the

hie

rarc

hy

con

cep

ts.

•N

eith

er c

an th

ey b

e co

nstr

ucte

d or

ver

ified

with

out t

he co

mp

ute

r to

ols

.

CP

-net

s ar

e of

ten

used

to

ge

the

r with

oth

ergr

aphi

cal d

escr

iptio

n la

ngua

ges,

suc

h as

SA

DT

,S

DL

and

bloc

k di

agra

ms.

•T

his

mea

ns th

at th

e us

er d

oes

not h

ave

to le

arn

aco

mpl

etel

y ne

w la

ng

ua

ge

.

CP

-net

s ar

e w

ell-s

uite

d fo

r verific

atio

n of

exi

stin

gde

sign

s –

in p

artic

ular

con

curr

ent s

yste

ms.

•C

P-n

ets

can

also

be

used

to

de

sig

n new

sys

tem

s.

•T

hen

it is

pos

sibl

e to

use

the insi

gh

t gai

ned

thro

ugh

the

mod

ellin

g, s

imul

atio

n an

dve

rific

atio

n ac

tiviti

es –

to im

pro

ve th

e de

sign

itsel

f.

Col

oure

d P

etri

Net

s 44

Par

t 3:

Con

stru

ctio

n an

dS

imul

atio

n of

CP

-net

sC

P-n

ets

have

an in

teg

rate

d set

of r

ob

ust c

ompu

ter

tool

s w

ith re

liab

le s

up

po

rt:

•C

on

stru

ctio

n and

mo

difi

catio

n of C

PN

mod

els.

•S

ynta

x ch

eck

ing (e.

g., t

ypes

and

mod

ule

inte

rfac

es).

•In

tera

ctiv

e s

imu

latio

n, e.g

., to

gai

n ad

ditio

nal

unde

rsta

ndin

g of

the

mod

elle

d sy

stem

. Can

als

obe

use

d fo

r de

bu

gg

ing.

•A

uto

ma

tic s

imu

latio

ns, e

.g.,

to o

btai

n pe

rfor

man

cem

easu

res.

Can

als

o be

use

d fo

r p

roto

typ

ing.

•V

erific

atio

n to

pro

ve b

ehav

iour

al p

rope

rtie

s.

–S

tate

sp

ace

s (al

so c

alle

d re

acha

bilit

y gr

aphs

and

occu

rren

ce g

raph

s).

–P

lace

inva

ria

nts.

The

com

pute

r to

ols

are

avai

labl

e on

:

•S

un S

parc

with

Sol

aris

.

•H

P w

ith H

PU

X

•In

tel P

Cs

with

Lin

ux.

•M

acin

tosh

with

Mac

OS

.

132

Col

oure

d P

etri

Net

s 45

CP

N e

dito

r

Eac

h pa

ge

is s

how

n in

its

own w

ind

ow

.

The

use

r pe

rfor

ms

an o

pera

tion

by s

elec

ting

ano

bje

ct a

nd a

com

ma

nd f

or it

, e.g

.:

•S

elec

t a tra

nsi

tion

(by

poin

ting

with

the

mou

se).

•S

elec

t the

des

ired co

mm

an

d (by

poi

ntin

g in

the

corr

espo

ndin

g dr

op-d

own

men

u).

Com

man

ds c

an b

e pe

rfor

med

on

a se

t o

f o

bje

cts.

Col

oure

d P

etri

Net

s 46

Edi

tor

know

s sy

ntax

of C

P-n

ets

Som

e ki

nds

of e

rror

s ar

e imp

oss

ible,

e.g

.:

•A

n ar

c be

twee

n two

pla

ces o

r tw

o t

ran

sitio

ns.

•A

pla

ce w

ith tw

o c

olo

ur

sets o

r an

arc

with

two

arc

exp

ress

ion

s.

•A

tran

sitio

n w

ith a

colo

ur

set.

•P

ort a

ssig

nmen

t inv

olvi

ng a

pla

ce w

hich

is a

no

n-s

ock

et or

a n

on

-po

rt.

•A

cyc

lic s

et o

f su

bst

itutio

n t

ran

sitio

ns.

The

edi

tor

beha

ves in

telli

ge

ntly

:

AU

BU

Tra

ns[x

<>

y]

AU

BU

Tra

ns[x

<>

y]

AU

BU

Tra

ns[x

<>

y]

AU

BU

(x,y

)

f(x,

y)

(x,y

)

f(x,

y)

(x,y

)

f(x,

y)

•W

hen

a no

de is

rep

osi

tion

ed o

r re

size

d the

surr

ound

ing

arcs

and

insc

riptio

ns a

rea

uto

ma

tica

lly a

dju

ste

d.

•W

hen

a no

de is

dele

ted t

he s

urro

undi

ng a

rcs

are

au

tom

atic

ally

de

lete

d.

133

Col

oure

d P

etri

Net

s 47

Attr

ibut

esE

ach

grap

hica

l obj

ect h

as it

s ow

n a

ttrib

ute

s.

The

y de

term

ine

how

the

obje

ct a

ppea

rs o

n th

esc

reen

/prin

t-ou

ts:

Tra

nsiti

on

TTTT Trrrr raaaa annnn nssss siiii itttt tiiii ioooo onnnn n

Tra

nsi

tio

n

• T

ext a

ttrib

utes

• G

raph

ical

attr

ibut

es

• Sh

ape

attr

ibut

es

Eac

h kin

d o

f o

bje

cts h

as it

s ow

n de

fau

lts:

Plac

e N

ame

Col

our

Set

Initi

al M

arki

ng

Tra

nsi

tio

nN

ame

[Gua

rd]

Arc

Exp

ress

ion

Def

aults

can

be c

ha

ng

ed a

nd th

ey c

an b

eo

verw

ritt

en

(for

indi

vidu

al o

bjec

ts).

Col

oure

d P

etri

Net

s 48

Eas

y to

exp

erim

ent

[not

(lock

ed(r

))]

C

Get

Rec

ord

Lock

@+

CP

Ulo

ck()

TR

AN

SP

7O

ut

TR

AN

S_R

EQ

P1

In

LOC

K_D

B

Lock

Tab

le

FG

TR

AN

S_R

EQ

P2

[i>1]

Res

erve

Pag

e

C @+

CP

Ure

serv

e()

AP

P_R

EQ

P4

AP

P_R

EQ

P6

Rea

d D

ata

Fro

m B

uffe

r @+

CP

Ure

ad()

C

[buf

fere

d(r)

]

Byp

ass

Dis

k A

cces

s

C

@+

CP

Uby

pass

()

Rec

ord

Rea

d

HS

Dis

kAcc

ess

AP

P_R

EQ

P5

C@

+C

PU

upda

te()

Upd

ate

Buf

fer

BU

FF

ER

_DBBuf

fer

Tab

le

FG

TR

AN

S_R

EQ

_QU

EU

E

( [ ]

, [ ]

)P

3

[nee

d_di

sk_r

ead(

r),

page

_cle

an_p

roc(

)]

Que

ue R

eque

st

C

In L

ine

Cle

anin

g

HS

Pag

eCle

anin

g

(t,r

)

(t,r

)

ldb

(TA

PP

t,dr

)

(TA

PP

t,D

iskR

eqO

fRec

(r))

(t,r

)

(TA

PP

t,D

iskR

eqO

fRec

(r))

t

i

(ap,

dr)

(ap,

dr)

i

tq

((t,r

)::tr

l1,tr

l2)

(t,r

)

tq(trl1

,trl2

)

i

i

i

i -1

Can

we

impr

ove

the la

you

t of t

his

page

?

134

Col

oure

d P

etri

Net

s 49

Impr

oved

layo

ut [not

(lock

ed(r

))]

C

Get

Rec

ord

Lock

@+

CP

Ulo

ck()

TR

AN

SP

7O

ut

TR

AN

S_R

EQ

P1

In

LOC

K_D

B

Lock

Tab

le

FG

TR

AN

S_R

EQ

P2

[i>1]

Res

erve

Pag

e

C@

+C

PU

rese

rve(

)

AP

P_R

EQ

P4

AP

P_R

EQ

P6

Rea

d D

ata

Fro

m B

uffe

r@

+C

PU

read

()C

[buf

fere

d(r)

]

Byp

ass

Dis

k A

cces

s

C

@+

CP

Uby

pass

()

Rec

ord

Rea

d

HS

Dis

kAcc

ess

AP

P_R

EQ

P5

C@

+C

PU

upda

te()

Upd

ate

Buf

fer

BU

FF

ER

_DB

Buf

fer

Tab

le

FG

TR

AN

S_R

EQ

_QU

EU

E

( [ ]

, [ ]

)P

3

[nee

d_di

sk_r

ead(

r),

page

_cle

an_p

roc(

)]

Que

ue R

eque

st

C

In L

ine

Cle

anin

g

HS

Pag

eCle

anin

g

(t,r

)

(t,r

)

ldb

(TA

PP

t,dr

)

(TA

PP

t,D

iskR

eqO

fRec

(r))

(t,r

)

(TA

PP

t,D

iskR

eqO

fRec

(r))

t

i

(ap,

dr)

(ap,

dr)

i

tq

((t,r

)::tr

l1,tr

l2)

(t,r

)

tq(trl1

,trl2

)

i

i

i

i -1

Col

oure

d P

etri

Net

s 50

How

to m

ake

a ne

w s

ubpa

ge

Rec

eive

all

Ack

now

ledg

-m

ents

Upd

ate

and

Sen

d M

essa

ges

Sen

d an

Ack

now

ledg

-m

ent

Rec

eive

aM

essa

ge

Per

form

ing

DB

M

Inac

tive

DB

M

DB

M

Wai

ting

DB

M

Unu

sed

ME

S

ME

S

Sen

t

ME

S

Rec

eive

d ME

S

Ack

now

ledg

ed

ME

S

Act

ive

E

Pas

sive

Ee

Dat

aBas

e#1

Mes

(s)

(s,r

)

(s,r

)

(s,r

)

(s,r

)M

es(s

)

Mes

(s)

Mes

(s)

ss s

s

rr r

r

e eee

We

wan

t to m

ove

the

four

sel

ecte

d no

des

to a

n

ew

pa

ge

– an

d re

plac

e th

em b

y a su

bst

itutio

ntr

an

sitio

n:

•T

his

is d

one

by a

sin

gle

com

man

d –

calle

d M

ove

to S

ub

pa

ge.

135

Col

oure

d P

etri

Net

s 51

Res

ult o

f Mov

e to

Sub

page

Rec

eive

all

Ack

now

ledg

-m

ents

Upd

ate

and

Sen

d M

essa

ges

Inac

tive

DB

M

DB

MW

aitin

g

DB

M

Unu

sed

ME

S

ME

S

Sen

t

ME

S

Ack

now

ledg

ed

ME

S

Act

ive

E

Pas

sive

Ee

HS

New

#2

Dat

aBas

e#1

Rec

eive

aM

essa

ge

Rec

eive

d ME

S

Sen

d an

Ack

now

ledg

-m

ent

Per

form

ing

DB

M

Inac

tive

DB

M

DB

M

I/O

Ack

now

ledg

ed

ME

S

Ou

t

Sen

t

ME

S

In

New

#2

Mes

(s)

(s,r

)

(s,r

)

Mes

(s)

Mes

(s)

Mes

(s)

ss s

s

r r

e eee

(s,r

)

(s,r

)

r r

(s,r

)

(s,r

)

r r

Col

oure

d P

etri

Net

s 52

Mov

e to

Sub

page

is c

ompl

exT

he M

ove

to

Su

bp

ag

e com

man

d is

com

ple

x. T

heco

mm

and:

•C

heck

s th

e leg

alit

y o

f th

e s

ele

ctio

n (al

l bor

der

node

s m

ust b

e tr

ansi

tions

).

•C

reat

es th

e ne

w p

ag

e.

•M

ove

s th

e s

ub

ne

t to

the

new

pag

e.

•P

rom

pts

th

e u

ser to

cre

ate

a ne

w tr

ansi

tion

whi

chbe

com

es th

e su

pern

ode

for

the

new

sub

page

.

•C

reat

es th

e po

rt p

lace

s by

copy

ing

thos

e pl

aces

whi

ch w

ere

next

to th

e se

lect

ed s

ubne

t.

•C

alcu

late

s th

e po

rt t

ype

s (in

, out

or

in/o

ut).

•C

reat

es th

e co

rres

pond

ing po

rt in

scrip

tion

s.

•C

onst

ruct

s th

e ne

cess

ary

arc

s be

twee

n th

e po

rtno

des

and

the

sele

cted

sub

net.

•D

raw

s th

e arc

s su

rrou

ndin

g th

e ne

w tr

ansi

tion.

•C

reat

es a

hie

rarc

hy

insc

rip

tion f

or th

e ne

wtr

ansi

tion.

•U

pdat

es th

e hie

rarc

hy

pa

ge

.

All

thes

e th

ings

are

don

e in

a

few

se

con

ds.

136

Col

oure

d P

etri

Net

s 53

Top

-dow

n an

d bo

ttom

-up

Mo

ve t

o S

ub

pa

ge sup

port

s top

-do

wn d

evel

opm

ent.

How

ever

, it i

s al

so p

ossi

ble

to w

ork bo

tto

m-u

p – o

rus

e a m

ixtu

re o

f top

-dow

n an

d bo

ttom

-up.

The

Su

bst

itutio

n T

ran

sitio

n com

man

d is

use

d to

rela

te a

sub

stitu

tion

tran

sitio

n to

an

exi

stin

g p

ag

e.

The

com

man

d:

•M

akes

the h

iera

rch

y p

ag

e a

ctiv

e.

•P

rom

pts

th

e u

ser to

sel

ect t

he d

esire

d su

bpag

e;w

hen

the

mou

se is

mov

ed o

ver

a pa

ge n

ode

itbl

inks

, unl

ess

it is

ille

gal (

beca

use

sele

ctin

g it

wou

ld m

ake

the

page

hie

rarc

hy c

yclic

).

•W

aits

unt

il a

blin

king

pa

ge

no

de h

as b

een

sele

cted

.

•T

ries

to d

educ

e th

e port

ass

ign

me

nt by

mea

ns o

fa

set o

f rul

es w

hich

look

s at

the

port

/soc

ket

nam

es a

nd th

e po

rt/s

ocke

t typ

es.

•C

reat

es th

e hie

rarc

hy

insc

rip

tion w

ith th

e na

me

and

num

ber

of th

e su

bpag

e an

d w

ith th

ose

part

sof

the

port

ass

ignm

ent w

hich

cou

ld b

eau

tom

atic

ally

ded

uced

.

•U

pdat

es th

e hie

rarc

hy

pa

ge

.

Col

oure

d P

etri

Net

s 54

Syn

tax

chec

king

Whe

n a

CP

N d

iagr

am h

as b

een

cons

truc

ted

it ca

nbe

syn

tax

che

cke

d.

The

mos

t com

mon

err

ors

are:

•S

ynta

x er

rors

in th

e de

cla

ratio

ns.

•S

ynta

x er

rors

in arc

exp

ress

ion

s o

r g

ua

rds.

•T

ype

mis

ma

tch b

etw

een

arc

expr

essi

ons

and

colo

ur s

ets.

Syn

tax

chec

king

is incr

em

en

tal:

•W

hen

a co

lour

set

, gua

rd o

r an

arc

exp

ress

ion

isch

ange

d, it

is su

ffic

ien

t to r

eche

ck th

e ne

are

stsu

rro

un

din

gs.

•A

nalo

gous

ly, i

f an a

rc is

add

ed o

r re

mov

ed.

All

CP

N d

iagr

ams

in th

is s

et o

f lec

ture

not

es a

rem

ade

by m

eans

of t

he C

PN

edi

tor.

137

Col

oure

d P

etri

Net

s 55

CP

N s

imul

ator

Whe

n a s

ynta

ctic

al c

orr

ect C

PN

dia

gram

has

bee

nco

nstr

ucte

d, th

e C

PN

tool

gen

erat

es th

e ne

cess

ary

cod

e t

o p

erf

orm

sim

ula

tion

s.

The

sim

ulat

ion

code

:

•C

alcu

late

s w

heth

er th

e in

divi

dual

tran

sitio

ns a

ndbi

ndin

gs a

re en

ab

led

.

•C

alcu

late

s th

e eff

ect

of

occ

urr

ing

tra

nsi

tion

s an

db

ind

ing

s.

The

cod

e ge

nera

tion

is

incr

em

en

tal. H

ence

it is

fast

to m

ake

smal

l cha

nges

to th

e C

PN

dia

gram

.

We

dist

ingu

ish

betw

een

two

kind

s of

sim

ulat

ions

:

•In

an

inte

ract

ive

sim

ulat

ion

the

user

is in

con

trol

,bu

t mos

t of t

he w

ork

is d

one

by th

e sy

stem

.

•In

an

au

tom

atic

sim

ulat

ion

the

syst

em d

oes

all

the

wor

k.

Col

oure

d P

etri

Net

s 56

Inte

ract

ive

sim

ulat

ion

Sen

dP

acke

tT

rans

mit

Pac

ket

Rec

eive

Pac

ket

Rec

eive

Ack

now

.T

rans

mit

Ack

now

.

Sen

d

INT

xDA

TA

8

1`(1

,"M

odel

lin")

+ 1

`(2,

"g a

nd A

n")

+ 1

`(3,

"aly

sis

b")

+ 1

`(4,

"y M

eans

")

+ 1

`(5,

"of C

olou

")+

1`(

6,"r

ed P

etr"

)+

1`(

7,"i

Net

s##"

)+

1`(

8,"#

####

###"

)

Nex

tSen

dIN

T

1

11`

5

DIN

T

11`

6

A

INT

xDA

TA

Rec

eive

dD

AT

A

""1

1`"M

odel

ling

and

Ana

lysi

s by

Mea

ns

of C

olou

"

Nex

tRec

INT

1

11`

6

B

INT

xDA

TA

11`

(5,"

of C

olou

")

CIN

T

11`

6

Sen

der

Net

wo

rkR

ecei

ver

RP

8

Int_

0_10

11`

8

RA

Int_

0_10

81

1`8

(n,p

)(n

,p)

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

(n,p

)

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

n

k

if n=

kth

en k

+1

else

k

if O

k(s,

r)th

en 1

`nel

se e

mpt

y

n

kn

n

str

ss

Sim

ula

tion

re

sults

are

show

n di

rect

ly o

n th

e C

P-n

et:

•T

he u

ser

can

see

the en

ab

led

tra

nsi

tion

s and

the

ma

rkin

gs o

f the

indi

vidu

al p

lace

s.

To

exe

cute

a s

tep, the

use

r:

•S

ele

cts o

ne o

f the

ena

bled

tran

sitio

ns.

•T

hen

he ei

the

r ent

ers

a bi

ndin

g or

ask

s th

esi

mul

ator

to c

alcu

late

all

the

enab

led

bind

ings

, so

that

he

can

sele

ct o

ne.

138

Col

oure

d P

etri

Net

s 57

Exe

cutio

n of

a s

tep

The

sim

ulat

or:

•C

heck

s th

e leg

alit

y a

nd

en

ab

ling o

f the

bin

ding

.

•C

alcu

late

s th

e resu

lt o

f th

e e

xecu

tion

.

The

use

r de

term

ine

s whe

ther

the

sim

ulat

ordi

spla

ys th

e to

kens

whi

ch a

re a

dded

/rem

oved

:

B

11`

(3,"

alys

is b

")

Rec

eive

Pac

ket

Nex

tRec

11`

3

Rec

eive

d1

1`"M

odel

ling

and

An"

C

(n,p

)1

1`(3

,"al

ysis

b")

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

11`

"Mod

ellin

g an

d A

naly

sis

b"

if n=

kth

en k

+1

else

k

11`

4

k

11`

3

if n=

kth

en k

+1

else

k

11`

4

str1

1`"M

odel

ling

and

An"

Col

oure

d P

etri

Net

s 58

Inte

ract

ive

sim

ulat

ion

with

rand

om s

elec

tion

of s

teps

The

sim

ulat

or ch

oo

ses b

etw

een

conf

lictin

gtr

ansi

tions

and

bin

ding

s (b

y m

eans

of a

ra

nd

om

nu

mb

er

ge

ne

rato

r).

•T

he u

ser

can o

bse

rve a

ll de

tails

, e.g

., th

e m

arki

ngs

the

enab

ling

and

the

adde

d/re

mov

ed to

kens

.

•T

he s

imul

ator

sho

ws

the

pa

ge on

whi

ch e

ach

step

is e

xecu

ted

– by

mov

ing

the

corr

espo

ndin

gw

indo

w to

the

top

of th

e sc

reen

.

•T

he u

ser

can

set bre

akp

oin

ts so

that

he

has

the

nece

ssar

y tim

e to

insp

ect m

arki

ngs,

ena

blin

gs,

etc.

A s

imul

atio

n w

ith th

is a

mou

nt o

f gra

phic

al fe

edba

ckis

slo

w (

typi

cally

a fe

w tr

ansi

tions

per

min

ute)

:

•It

take

s a

lot o

f tim

e to

upd

ate

the

grap

hics

.

•A

use

r ha

s no

cha

nce

to fo

llow

a fa

st s

imul

atio

n.

It is

pos

sibl

e to

turn

off

sele

cted

par

ts o

f the

gra

ph

ica

l fe

ed

ba

ck, e

.g.:

•A

dded

and

rem

oved

to

ken

s.

•O

bser

vatio

n of

unin

tere

stin

g p

ag

es.

139

Col

oure

d P

etri

Net

s 59

Aut

omat

ic s

imul

atio

nT

he s

imul

ator

cho

ose

s bet

wee

n co

nflic

ting

tran

sitio

ns a

nd b

indi

ngs

(by

mea

ns o

f a

ran

do

mn

um

be

r g

en

era

tor).

The

use

r do

es

no

t int

end

to fo

llow

the

sim

ulat

ion:

•T

he s

imul

atio

n ca

n be

ve

ry f

ast

– se

vera

lhu

ndre

d st

eps

per

seco

nd.

•T

he u

ser

spec

ifies

som

e sto

p c

rite

ria,

whi

chde

term

ine

the

dura

tion

of th

e si

mul

atio

n.

•W

hen

the

sim

ulat

ion

stop

s th

e gr

aphi

cs o

f the

CP

-net

is up

da

ted

.

•T

hen

the

user

can

insp

ect a

ll de

tails

of t

hegr

aphi

cs, e

.g.,

the en

ab

ling a

nd th

e ma

rkin

g.

•A

utom

atic

sim

ulat

ions

can

be mix

ed w

ithin

tera

ctiv

e si

mul

atio

ns.

To

find

out w

hat h

appe

ns d

urin

g an

a

uto

ma

ticsi

mu

latio

n the

use

r ha

s a

num

ber

of c

hoic

es.

Col

oure

d P

etri

Net

s 60

Sim

ulat

ion

repo

rt1

Sen

dPac

k@(1

:Top

#1)

{n =

1, p

= "

Mod

ellin

"}2

Tra

nPac

k@(1

:Top

#1)

{n =

1, p

= "

Mod

ellin

", r

= 6

, s =

8}

3S

endP

ack@

(1:T

op#1

){n

= 1

, p =

"M

odel

lin"}

4T

ranP

ack@

(1:T

op#1

){n

= 1

, p =

"M

odel

lin",

r =

3, s

= 8

}5

Rec

Pac

k@(1

:Top

#1)

{k =

1, n

= 1

, p =

"M

odel

lin",

str

= "

"}6

Sen

dPac

k@(1

:Top

#1)

{n =

1, p

= "

Mod

ellin

"}7

Sen

dPac

k@(1

:Top

#1)

{n =

1, p

= "

Mod

ellin

"}8

Tra

nAck

@(1

:Top

#1)

{n =

2, r

= 2

, s =

8}

9T

ranP

ack@

(1:T

op#1

){n

= 1

, p =

"M

odel

lin",

r =

7, s

= 8

}10

Rec

Pac

k@(1

:Top

#1)

{k =

2, n

= 1

, p =

"M

odel

lin",

str

= "

Mod

ellin

"}11

Rec

Ack

@(1

:Top

#1)

{k =

1, n

= 2

}12

Rec

Pac

k@(1

:Top

#1)

{k =

2, n

= 1

, p =

"M

odel

lin",

str

= "

Mod

ellin

"}13

Tra

nAck

@(1

:Top

#1)

{n =

2, r

= 7

, s =

8}

14T

ranP

ack@

(1:T

op#1

){n

= 1

, p =

"M

odel

lin",

r =

6, s

= 8

}15

Rec

Ack

@(1

:Top

#1)

{k =

2, n

= 2

}16

Sen

dPac

k@(1

:Top

#1)

{n =

2, p

= "

g an

d A

n"}

17T

ranA

ck@

(1:T

op#1

){n

= 2

, r =

6, s

= 8

}18

Rec

Pac

k@(1

:Top

#1)

{k =

2, n

= 1

, p =

"M

odel

lin",

str

= "

Mod

ellin

"}19

Rec

Ack

@(1

:Top

#1)

{k =

2, n

= 2

}20

Sen

dPac

k@(1

:Top

#1)

{n =

2, p

= "

g an

d A

n"}

The

sim

ula

tion

re

po

rt sho

ws

the tr

an

sitio

ns w

hich

have

occ

urre

d. T

he u

ser

dete

rmin

es w

heth

er h

eal

so w

ants

to s

ee th

e bin

din

gs.

140

Col

oure

d P

etri

Net

s 61

Cha

rts

Pac

kets

Rec

eive

d

Step

No.

020

4060

8010

012

014

016

018

020

0

Pack

et N

o

012345678910

Pac

kets

pack

1

pack

2

pack

3

pack

4

pack

5

pack

6

pack

7

pack

8

5 5 7 5 4

01

23

45

67

89

10

01

23

45

67

89

10

Suc

cess

es

Fai

lure

s

Los

t

Enr

oute

The

se c

hart

s ar

e us

ed to

sho

w th

e p

rog

ress

of a

sim

ulat

ion

of th

e si

mpl

e pr

otoc

ol:

•T

he u

pper

cha

rt is

upd

ated

eac

h tim

e a

new

pack

et is

succ

ess

fully

re

ceiv

ed

.

•T

he lo

wer

cha

rt is

upd

ated

for

ea

ch 5

0 s

tep

s.

Col

oure

d P

etri

Net

s 62

Oth

er k

inds

of g

raph

ics

u(1)

Sho

rt

u(3)

Sho

rt

u(4)

Inac

u(6)

Rin

g

u(10

)

NoT

o

u(9)

Inac

u(8)

Con

n

u(7)

Con

n

u(5)

Sho

rt

u(2)

Long

Thi

s gr

aphi

c is

use

d to

dis

play

the

stat

e of

a

sim

ple

tele

ph

on

e s

yste

m.

The

gra

phic

s is

upd

ated

eac

htim

e on

e of

the

tele

phon

es c

hang

es to

a n

ew s

tate

:

•T

elep

hone

s u(

7) a

nd u

(8)

are

con

ne

cte

d.

•T

elep

hone

u(2

) is

cal

ling

u(6)

whi

ch is

rin

gin

g.

•T

elep

hone

u(1

0) is

cal

ling

u(2)

. Thi

s ca

ll w

ill

no

tsu

cce

ed b

ecau

se u

(2)

alre

ady

is e

ngag

ed.

141

Col

oure

d P

etri

Net

s 63

Cod

e se

gmen

tsE

ach

tran

sitio

n m

ay h

ave

a co

de s

egm

ent,

i.e.,

ase

quen

ce o

f pro

gra

m in

stru

ctio

ns w

hich

are

exec

uted

eac

h tim

e th

e tr

ansi

tion

occu

rs.

NoT

one

U

Long

U

Fre

eC

input (x,y);

action

UpdateState(x,Long);

UpdateState(y,Ringing);

UpdateConn(x,y,Call);

Rin

ging

U

Req

uest

UxU

Cal

lU

xU

Eng

aged

U

Inac

tive

U

xx

y

(x,y

)

(x,y

)

y

y

•T

he in

stru

ctio

ns in

cod

e se

gmen

t are

use

d to

up

da

te c

ha

rts

an

d g

rap

hic

s.

•T

his

is d

one

by c

allin

g a

num

ber

of

libra

ryfu

nct

ion

s.

•U

sual

ly, t

he c

ode

segm

ent d

oes

no

t inf

luen

ce th

eb

eh

avi

ou

r of t

he C

P-n

et (

i.e.,

the

enab

ling

and

occu

rren

ce).

•H

owev

er, a

cod

e se

gmen

t may

re

ad

an

d w

rite

from

file

s.

•In

this

way

it is

pos

sibl

e to

in

pu

t va

lue

s to

beus

ed d

urin

g th

e si

mul

atio

n, o

r to

o

utp

ut

sim

ula

tion

re

sults

.

Col

oure

d P

etri

Net

s 64

Sta

ndar

d M

LD

ecla

ratio

ns, n

et in

scrip

tions

and

cod

e se

gmen

tsar

e sp

ecifi

ed in

a pro

gra

mm

ing

lan

gu

ag

e cal

led

Sta

nd

ard

ML.

•S

tro

ng

ly t

ype

d, fu

nct

ion

al la

ngua

ge.

•D

ata

typ

es c

an b

e:–

Ato

mic

(inte

gers

, rea

ls, s

trin

gs, b

oole

ans

and

enum

erat

ions

).–

Str

uct

ure

d (pr

oduc

ts, r

ecor

ds, u

nion

s, li

sts

and

subs

ets)

.

•A

rbitr

ary

com

plex

fun

ctio

ns a

nd o

pe

ratio

ns c

anbe

def

ined

(po

lym

orph

ism

and

ove

rload

ing)

.

•C

ompu

tatio

nal p

ower

of e

xpre

ssio

ns a

reeq

uiva

lent

to la

mb

da

ca

lcu

lus (an

d he

nce

toT

urin

g m

achi

nes)

.

•D

evel

oped

at E

din

bu

rgh

Un

ive

rsity

by R

obin

Miln

er a

nd h

is g

roup

.

•S

tand

ard

ML

is w

ell-k

now

n, w

ell-t

este

d an

dve

ry g

ener

al. S

ever

al

text

bo

oks

are

avai

labl

e.

142

Col

oure

d P

etri

Net

s 65

Tim

e an

alys

isC

P-n

ets

can

be e

xten

ded

with

a

time

co

nce

pt.

Thi

s m

eans

that

the sa

me

lan

gu

ag

e can

be

used

toin

vest

igat

e:

•L

og

ica

l co

rre

ctn

ess

.D

esire

d fu

nctio

nalit

y, a

bsen

ce o

f dea

dloc

ks, e

tc.

•P

erf

orm

an

ce.

Rem

ove

bottl

enec

ks. P

redi

ct m

ean

wai

ting

times

and

aver

age

thro

ughp

ut. C

ompa

re d

iffer

ent

stra

tegi

es.

In a

tim

ed C

P-n

et e

ach

toke

n ca

rrie

s a

colo

ur (

data

valu

e) a

nd a

time

sta

mp (

telli

ng w

hen

it ca

n be

used

).

Tim

e st

amps

are

spe

cifie

d by

exp

ress

ions

:

•T

ime

stam

ps c

an d

epen

d up

on

colo

ur

valu

es.

•T

ime

stam

ps c

an b

e sp

ecifi

ed b

y p

rob

ab

ility

dis

trib

utio

ns.

•T

his

mea

ns th

at w

e, e

.g.,

can

spec

ify

fixe

dde

lays

, inte

rva

l del

ays

and e

xpo

ne

ntia

l del

ays.

Col

oure

d P

etri

Net

s 66

A ti

med

CP

-net

for

prot

ocol

Sen

dP

acke

t @+

9

Tra

nsm

itP

acke

t @+

Del

ay()

Rec

eive

Pac

ket

@+

17

Rec

eive

Ack

now

.

@+

7

Tra

nsm

itA

ckno

w.

@+

Del

ay()

Sen

d

INT

xDA

TA

Nex

tSen

dIN

T

1

DIN

TA

INT

xDA

TA

Rec

eive

dD

AT

A

""

Nex

tRec

INT

1

B

INT

xDA

TA

CIN

T

Sen

der

Net

wo

rkR

ecei

ver

RP

8

Int_

0_10

RA

Int_

0_10

8

Wai

t

TIM

E

100

(n,p

)(n

,p)

if O

K(s

,r)

then

1`(

n,p)

else

em

pty

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

kn

k

if n=

kth

en k

+1

else

k

if O

K(s

,r)

then

1`n

else

em

pty

n

kn

n

str

(n,p

)(n

,p)@

+w

ait

ss

wai

t

•F

or th

e th

ree S

en

d and

Re

ceiv

e ope

ratio

ns w

esp

ecify

a fix

ed

de

lay.

•F

or th

e ne

two

rk w

e sp

ecify

an in

terv

al d

ela

y, i.e

.,ra

ndom

del

ay b

etw

een

25 a

nd 7

5 tim

e un

its.

•T

he to

ken

colo

ur o

n pl

ace W

ait

spec

ifies

the

dela

y be

twee

n tw

o retr

an

smis

sio

ns o

f the

sam

epa

cket

.

The

com

pute

r to

ols

for

CP

-net

s al

so s

uppo

rtsi

mul

atio

n of

time

d C

P-n

ets.

143

Col

oure

d P

etri

Net

s 67

Tim

ed s

imul

atio

n of

pro

toco

lT

ime:

570

Sen

dP

acke

t @+

9

Tra

nsm

itP

acke

t @+

Del

ay()

Rec

eive

Pac

ket

@+

17

Rec

eive

Ack

now

.

@+

7

Tra

nsm

itA

ckno

w.

@+

Del

ay()

Sen

d

INT

xDA

TA

81`

(1,"

Mod

ellin

")@

[218

]+

1`(2

,"g

and

An"

)@[6

70]+

1`

(3,"

alys

is b

")@

[0]+

1`

(4,"

y M

eans

")@

[0]+

1`

(5,"

of C

olou

")@

[0]+

1`

(6,"

red

Pet

r")@

[0]+

1`

(7,"

i Net

s##"

)@[0

]+

1`(8

,"##

####

##")

@[0

]

Nex

tSen

dIN

T

1

11`

2@[5

70] D

INT

11`

3@[5

93]

A

INT

xDA

TA

1

1`(2

,"g

and

An"

) @

[570

]

Rec

eive

dD

AT

A

11`

"Mod

ellin

g an

d A

n"

""

Nex

tRec

INT

1

11`

3@[5

48]B

INT

xDA

TA

CIN

T

Sen

der

Net

wo

rkR

ecei

ver

RP

8

Int_

0_10

11`

8

RA

Int_

0_10

11`

88

Wai

tT

IME

11`

100

(n,p

)(n

,p)

if O

K(s

,r)

then

1`(

n,p)

else

em

pty

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

kn

k

if n=

kth

en k

+1

else

k

if O

K(s

,r)

then

1`n

else

em

pty

n

kn

n

str

(n,p

)(n

,p)@

+w

ait

ss

wai

t

•M

odel

tim

e is

now

570

.

•S

en

d P

ack

et has

sen

t a c

opy

of p

acke

t no.

2 a

ttim

e 5

70

.

•If

no a

ckno

wle

dgem

ent a

rriv

es

an

oth

er

cop

y of

pack

et n

o. 2

will

be

sent

at tim

e 6

70

.

•T

he o

nly

tran

sitio

n w

hich

is e

nabl

ed a

t tim

e 57

0is

Tra

nsm

it P

ack

et.

Col

oure

d P

etri

Net

s 68

Tim

ed s

imul

atio

nsT

ime

d s

imu

latio

ns h

ave

the s

am

e f

aci

litie

s as

untim

ed s

imul

atio

ns, e

.g.:

•W

e ca

n sw

itch

betw

een in

tera

ctiv

e an

da

uto

ma

tic s

imul

atio

n.

•S

imu

latio

n r

ep

ort

s tell

the

time

at w

hich

the

indi

vidu

al tr

ansi

tions

occ

urre

d.

•W

e ca

n us

e ch

art

s and

oth

er k

inds

of re

po

rtin

gfa

cilit

ies.

It is

eas

y to

switc

h be

twee

n a ti

me

d an

d an

untim

ed

sim

ula

tion

.

144

Col

oure

d P

etri

Net

s 69

Cha

rts

for

a tim

ed s

imul

atio

n

Tim

e

Tim

e0

50

01

00

01500

2000

25

00

3000

3500

40

00

Ru

n N

o.

123456789

Ste

p

Ste

p0

20

40

60

80

100

12

0140

16

0

Run

No.

123456789

25 :

1080

100 :

1576

200 :

2014

25 :

119

100 :

78

200 :

53

•S

ho

rt in

terv

al b

etw

een

retr

ansm

isio

ns im

plie

sfa

st t

ran

smis

sio

n with

he

avy

use

of

the

ne

two

rk.

•L

on

g in

terv

al b

etw

een

retr

ansm

isio

ns im

plie

ssl

ow

tra

nsm

issi

on w

ith le

ss u

se o

f th

e n

etw

ork

.

•T

o ge

t re

liab

le r

esu

lts it

is n

eces

sary

to m

ake

ala

rge

nu

mb

er o

f le

ng

thy s

imul

atio

n ru

ns.

Col

oure

d P

etri

Net

s 70

Par

t 4: V

erifi

catio

n o

f CP

-net

sIn

this

par

t of t

he ta

lk w

e de

scrib

e th

e tw

o m

ost

impo

rtan

t met

hods

for ve

rific

atio

n of

CP

-net

s:

•S

tate

sp

ace

s (al

so c

alle

d re

acha

bilit

y gr

aphs

and

occu

rren

ce g

raph

s).

•P

lace

inva

ria

nts.

We

also

des

crib

e ho

w th

e ve

rific

atio

n m

etho

ds a

resu

ppor

ted

by co

mp

ute

r to

ols.

145

Col

oure

d P

etri

Net

s 71

Sta

te s

pace

ana

lysi

s

AU3`q

BU2`p

CU

DU

EUT

1

T2

T3

T4

T5

R E

1ètype

U =

with

p |

q;

type

E =

with

e;

var

x : U

;

S E

3è

T E

2è

x x x x x x x x x

if x=

qth

en 1

`qel

se e

mpt

y

e

if x=

q th

en 1

èel

se e

mpt

y

case

x o

f p

=>

2è

| q =

> 1

è

2è

e

if x=

p th

en 1

èel

se e

mpt

y

e

case

x o

f p

=>

2è

| q =

> 1

è

if x=

pth

en 1

`pel

se e

mpt

y

To

obta

in a

finite

sta

te s

pace

we

rem

ove

the

cycl

eco

unte

rs. O

ther

wis

e th

ere

wou

ld b

e an

in

finite

num

ber

of r

each

able

mar

king

s.

Col

oure

d P

etri

Net

s 72

Sta

te s

pace

for

reso

urce

allo

catio

n

0 2 2

QQ

PP

Q – – –

#5

1 3 2

QQ

QP

P – – –

#1

1 1 2

QQ

QP P – –

#2

0 1 2

QQ

PP Q – –

#9 1 1 2

QQ

PP – Q –

#10

1 1 1

QQ

PP – – Q

#12

1 1 1

QQ

QP – P –

#3

1 1 0

QQ

QP – – P

#4

0 0 2

QQ

PQ P – –

#6

0 0 1

QQ

PQ – P –

#7

0 0 0

QQ

PQ – – P

#8

0 0 2

QP

PQ – Q –

#11

0 0 1

QP

PQ – – Q

#13

(T1,

Q)

(T2,

P)

(T2,

Q) (T

3,Q

)

(T4,

Q)

(T5,

Q)

(T3,

P)

(T4,

P)

(T1,

Q)

(T2,

P)

(T3,

P)

(T5,

P)

(T1,

Q)

(T1,

Q)

(T4,

P)

(T5,

P)

(T1,

Q)

(T4,

Q)

(T1,

Q)

(T5,

Q)

Dire

cte

d g

rap

h with

:

•A

no

de f

or e

ach r

ea

cha

ble

ma

rkin

g (i.e

., st

ate)

.

•A

n a

rc fo

r ea

ch oc

curr

ing

bin

din

g e

lem

en

t.

146

Col

oure

d P

etri

Net

s 73

Som

e qu

estio

ns th

at c

an b

ean

swer

ed fr

om s

tate

spa

ces

Bo

un

de

dn

ess p

rope

rtie

s:

•W

hat i

s th

e ma

xim

al n

um

be

r of t

oken

s on

the

diffe

rent

pla

ces?

•W

hat i

s th

e min

ima

l nu

mb

er o

f tok

ens

on th

edi

ffere

nt p

lace

s?

•W

hat a

re th

e po

ssib

le t

oke

n c

olo

urs?

Ho

me p

rope

rtie

s:

•Is

it a

lwa

ys p

ossi

ble

to re

turn

to th

e in

itial

mar

king

?

Liv

en

ess

prop

ertie

s:

•A

re a

ll tr

ansi

tions

live

, i.e

., ca

n th

ey

alw

ays

beco

me

enab

led ag

ain

?

Col

oure

d P

etri

Net

s 74

Sta

te s

pace

rep

ort f

or r

esou

rce

allo

catio

n sy

stem

Sta

tistic

sO

ccur

renc

e G

raph

Nod

es:

13A

rcs:

20S

ecs:

1S

tatu

s:F

ull

Scc

Gra

phN

odes

:1

Arc

s:0

Sec

s:1

Bou

nded

ness

Pro

pert

ies

Upp

er In

tege

r B

ound

sA

:3

B:

3C

:1

D:

1E

:1

R:

1S

:3

T:

2

Upp

er M

ulti-

set B

ound

sA

:3`

qB

:2`

p+ 1

`qC

:1`

p+ 1

`qD

:1`

p+ 1

`qE

:1`

p+ 1

`qR

:1`

eS

:3`

eT

:2`

e

Low

er In

tege

r B

ound

sA

:1

B:

1C

:0

D:

0E

:0

R:

0S

:0

T:

0

Low

er M

ulti-

set B

ound

sA

:1`

qB

:1`

pC

:em

pty

D:

empt

yE

:em

pty

R:

empt

yS

:em

pty

T:

empt

y

147

Col

oure

d P

etri

Net

s 75

Sta

te s

pace

rep

ort (

cont

inue

d)H

ome

Pro

pert

ies

Hom

e M

arki

ngs:

All

Live

ness

Pro

pert

ies

Dea

d M

arki

ngs:

Non

e

Live

Tra

nsiti

ons:

All

Fai

rnes

s P

rope

rtie

s

T1:

No

Fai

rnes

sT

2:Im

part

ial

T3:

Impa

rtia

lT

4:Im

part

ial

T5:

Impa

rtia

l

Gen

erat

ion

of th

e st

ate

spac

e re

port

take

s on

ly a

few

se

con

ds.

•T

he r

epor

t con

tain

s a

lot o

f use

ful i

nfo

rma

tion

abou

t the

beh

avi

ou

r of t

he C

P-n

et.

•T

he r

epor

t is

exce

llent

for loca

ting

err

ors

or to

incr

ea

se o

ur

con

fide

nce i

n th

e co

rrec

tnes

s of

the

syst

em.

Col

oure

d P

etri

Net

s 76

Str

ongl

y co

nnec

ted

com

pone

nts

•S

ubgr

aph

whe

re all n

od

es

are

re

ach

ab

le f

rom

ea

ch o

the

r.

•M

axi

ma

l sub

grap

h w

ith th

is p

rope

rty.

148

Col

oure

d P

etri

Net

s 77

Str

ongl

y co

nnec

ted

com

pone

nts

are

very

use

ful

The

re a

re o

ften m

uch

fe

we

r str

ongl

y co

nnec

ted

com

pone

nts

than

nod

es:

•A

cyc

lic s

yste

m has

onl

y on

e st

rong

ly c

onne

cted

com

pone

nt.

•T

his

is, e

.g.,

the

case

for

the

reso

urce

allo

catio

nsy

stem

.

•T

he s

tro

ng

ly c

on

ne

cte

d c

om

po

ne

nts

can

be

dete

rmin

ed in

line

ar

time,

e.g

., by

Tar

jan’

sal

gorit

hm.

Str

ongl

y co

nnec

ted

com

pone

nts

can

be u

sed

toan

swer

que

stio

ns a

bout

h

om

e p

rop

ert

ies a

ndliv

en

ess

pro

pe

rtie

s.

Col

oure

d P

etri

Net

s 78

Sta

te s

pace

for

sim

ple

prot

ocol

Sen

dP

acke

tT

rans

mit

Pac

ket

Rec

eive

Pac

ket

Rec

eive

Ack

now

.T

rans

mit

Ack

now

.

Sen

d

INT

xDA

TA

1`(1

,"C

olou

red"

)+1`

(2,"

Pet

ri N

")+

1`(3

,"et

s###

##")

+1`

(4,"

####

####

")

Nex

tSen

dIN

T

1

DIN

TA

INT

xDA

TA

Rec

eive

dD

AT

A

""

Nex

tRec

INT

1

B

INT

xDA

TA

CIN

T

Sen

der

Net

wo

rkR

ecei

ver

RP

1

Int_

0_2

RA

Int_

0_21

Max

2 t

oke

ns

Max

1 t

oke

nM

ax 2

to

ken

s

Max

1 t

oke

n

(n,p

)(n

,p)

if O

k(s,

r)th

en 1

`(n,

p)el

se e

mpt

y

(n,p

)

(n,p

)

if n=

kan

dals

op<

>st

opth

en s

tr^p

else

str

if n=

kth

en k

+1

else

k

n

k

if n=

kth

en k

+1

else

k

if O

k(s,

r)th

en 1

`nel

se e

mpt

y

n

kn

n

str

ss

To

obta

in a

finite

sta

te s

pace

we

limit

the

num

ber

of to

kens

on

the

“buf

fer”

pla

ces

A, B

, C a

nd D

.O

ther

wis

e th

ere

wou

ld b

e an

in

finite

num

ber

ofre

acha

ble

mar

king

s.

Mor

eove

r, w

e no

w o

nly

have

4

pa

cke

ts and

ab

ina

ry c

ho

ice b

etw

een

succ

ess

and

failu

re.

149

Col

oure

d P

etri

Net

s 79

Sta

te s

pace

rep

ort f

or p

roto

col

Sta

tistic

sO

ccur

renc

e G

raph

Nod

es:

4298

Arc

s:15

887

Sec

s:53

Sta

tus:

Ful

l

Scc

Gra

phN

odes

:24

06A

rcs:

1167

7S

ecs:

17

Bou

nded

ness

Pro

pert

ies

Upp

er In

tege

r B

ound

sA

:1

B:

2C

:1

D:

2N

extR

ec:

1N

extS

end:

1R

A:

1R

P:

1R

ecei

ved:

1S

end:

4

Low

er In

tege

r B

ound

sA

:0

B:

0C

:0

D:

0N

extR

ec:

1N

extS

end:

1R

A:

1R

P:

1R

ecei

ved:

1S

end:

4

Col

oure

d P

etri

Net

s 80

Sta

te s

pace

rep

ort (

cont

inue

d)U

pper

Mul

ti-se

t Bou

nds

A:

1`(1

,"C

olou

red"

)+ 1

`(2,

" P

etri

N")

+1`

(3,"

ets#

####

")+

1`(

4,"#

####

###"

)B

:2`

(1,"

Col

oure

d")+

2`(

2,"

Pet

ri N

")+

2`(3

,"et

s###

##")

+ 2

`(4,

"###

####

#")

C:

1`2+

1`3

+ 1

`4+

1`5

D:

2`2+

2`3

+ 2

`4+

2`5

Nex

tRec

:1`

1+1`

2+1`

3+1`

4+1`

5N

extS

end:

1`1+

1`2+

1`3+

1`4+

1`5

RA

:1`

1R

P:

1`1

Rec

eive

d:1`

""+

1`"

Col

oure

d"+

1`"

Col

oure

d P

etri

N"+

1`"C

olou

red

Pet

ri N

ets#

####

"S

end:

1`(1

,"C

olou

red"

)+ 1

`(2,

" P

etri

N")

+1`

(3,"

ets#

####

")+

1`(

4,"#

####

###"

)

Low

er M

ulti-

set B

ound

sA

:em

pty

B:

empt

yC

:em

pty

D:

empt

yN

extR

ec:

empt

yN

extS

end:

empt

yR

A:

1`1

RP

:1`

1R

ecei

ved:

empt

yS

end:

1`(1

,"C

olou

red"

)+ 1

`(2,

" P

etri

N")

+1`

(3,"

ets#

####

")+

1`(

4,"#

####

###"

)

150

Col

oure

d P

etri

Net

s 81

Sta

te s

pace

rep

ort (

cont

inue

d)H

ome

Pro

pert

ies

Hom

e M

arki

ngs:

1[4

52]

Live

ness

Pro

pert

ies

Dea

d M

arki

ngs:

1[4

52]

Live

Tra

nsiti

ons:

Non

e

Fai

rnes

s P

rope

rtie

s

Sen

d P

acke

t:Im

part

ial

Tra

nsm

it P

acke

t:Im

part

ial

Rec

eive

Pac

ket:

No

Fai

rnes

sT

rans

mit

Ack

now

:N

o F

airn

ess

Rec

eive

Ack

now

:N

o F

airn

ess

Gen

erat

ion

of th

e st

ate

spac

e re

port

take

s on

ly a

few

se

con

ds.

•T

he r

epor

t con

tain

s a

lot o

f use

ful i

nfo

rma

tion

abou

t the

beh

avi

ou

r of t

he C

P-n

et.

•T

he r

epor

t is

exce

llent

for loca

ting

err

ors

or to

incr

ea

se o

ur

con

fide

nce i

n th

e co

rrec

tnes

s of

the

syst

em.

Col

oure

d P

etri

Net

s 82

Inve

stig

atio

n of

dea

d m

arki

ngW

e as

k th

e sy

stem

to d

ispl

ay m

arki

ng n

umbe

r 45

2.

45

2

8:0

452

NextSend = 5

NextRec = 5

Received = "Coloured Petri Nets#####"

Mar

king

no.

452

is th

e de

sire

d f

ina

l ma

rkin

g (al

lpa

cket

s ha

s be

en r

ecei

ved

in th

e co

rrec

t ord

er)

Mar

king

452

is de

ad

:

•T

his

impl

ies

that

the

prot

ocol

is

pa

rtia

lly c

orr

ect

(if e

xecu

tion

stop

s it

stop

s in

the

desi

red

final

mar

king

).

Mar

king

452

is a

hom

e m

ark

ing

:

•T

his

impl

ies

that

we a

lwa

ys h

ave

a c

ha

nce

to

finis

h c

orr

ect

ly (it

is im

poss

ible

to r

each

a s

tate

from

whi

ch w

e ca

nnot

rea

ch th

e de

sire

d fin

alm

arki

ng).

151

Col

oure

d P

etri

Net

s 83

Inve

stig

atio

n of

sho

rtes

t pat

hW

e as

k th

e sy

stem

to c

alcu

late

one

of t

he

sho

rte

stp

ath

s fro

m th

e in

itial

mar

king

to th

e de

ad m

arki

ng:

val path =

NodesInPath(1,452);

> val path =

[1,2,3,5,8,11,15,20,27,38,50,

64,80,102,133,164,199,243,

301,375,452] : Node list

Length(path);

> 20 : int

The

cal

cula

ted

path

con

tain

s 2

0 t

ran

sitio

ns.

•T

his

is a

s ex

pect

ed b

ecau

se th

ere

are

4 p

ack

ets

whi

ch e

ach

need

s 5 tr

an

sitio

ns t

o oc

cur.

Col

oure

d P

etri

Net

s 84

Dra

win

g of

sho

rtes

t pat

hW

e as

k th

e sy

stem

to d

raw

the

first

six

no

de

s in th

eca

lcul

ated

sho

rtes

t pat

h:

DisplayNodePath; [1,2,3,5,8,11];

> () : unit

1 1:1

1 NextSend = 1

NextRec = 1

Received = ""

2 1:2

2 NextSend = 1

NextRec = 1

Received = ""

A = 1`(1,"Coloured")

3 2:2

3 NextSend = 1`1

NextRec = 1`1

Received = ""

B = 1`(1,"Coloured")

5 3:3

5 NextSend = 1

NextRec = 2

Received = "Coloured"

C = 1`2

8 3:2

8 NextSend = 1

NextRec = 2


D = 1`2

11

5:1

11

NextSend = 2

NextRec = 2


1->2

SendPack

{p="Coloured",n=1}

2->3

TranPack

{s=1,r=1,p="Coloured",n=1}

3->5

RecPack

{str="",p="Coloured",n=1,k=1}

5->8

TranAck

{s=1,r=1,n=2}

8->11

RecAck

{n=2,k=1}

152

Col

oure

d P

etri

Net

s 85

Dra

w s

ubgr

aph

1 1:1

NextSend = 1

NextRec = 1

Received = ""

2 1:2

NextSend = 1

NextRec = 1

Received = ""


3 2:2

NextSend = 1

NextRec = 1

Received = ""


4 1:3

5 3:3

NextSend = 1

NextRec = 2


C = 1`2

9 2:2

6 3:4

7 2:1

8 3:2

NextSend = 1

NextRec = 2


D = 1`2

14

1:2

NextSend = 1

NextRec = 1

Received = ""



13

4:3

12

2:2

10

3:3

NextSend = 1

NextRec = 2



D = 1`2

11

5:1

SendPack

LoosePack

TranPack

SendPack

RecPack

LoosePack

TranPack

RecPack

LooseAck

TranAck

SendPack RecPack

LoosePack

SendPack

TranPack

LooseAck

TranAck

SendPack

SendPack

RecAck

Col

oure

d P

etri

Net

s 86

Non

-sta

ndar

d qu

estio

nsW

e as

k th

e sy

stem

to s

earc

h a

ll a

rcs

in th

e e

ntir

eg

rap

h an

d re

turn

the fi

rst

10

arc

s whe

re N

ext

Se

nd

has

a la

rge

r val

ue in

the s

ou

rce

ma

rkin

g than

it h

asin

the

de

stin

atio

n m

ark

ing

.

PredArcs

(EntireGraph,

fn a => ((ms_to_col(Mark.NextSend 1

(SourceNode a))) >

(ms_to_col(Mark.NextSend 1

(DestNode a)))),

10)

end;

>[10179,10167,10165,10159,10055,10052,10035,

10031,10019,10007] : Arc list

28

06

1

:4

NextSend = 4

NextRec = 5

Received = "Coloured Petri

Nets#####"

A = 1`(4,"########")

B = 2`(4,"########")

C = 1`5

D = 1`2+ 1`5

30

75

1

:4

NextSend = 2

NextRec = 5

Received = "Coloured

Petri Nets#####"

A = 1`(4,"########")

B = 2`(4,"########")

C = 1`5

D = 1`5

10179:2806->3075

RecAck = {n=2,k=4}

153

Col

oure

d P

etri

Net

s 87

Tem

pora

l log

icIt

is a

lso

poss

ible

to m

ake

ques

tions

by

mea

ns o

f aC

TL-

like

tem

po

ral l

og

ic.

Usu

ally

CT

L fo

cuse

s on

que

ries

abou

t st

ate

pro

pe

rtie

s, e.

g.:

•In

v(P

os(M

))ch

ecks

whe

ther

M is

a

ho

me

ma

rkin

g.

•E

v(de

ad)

chec

ks w

heth

er th

ere

are

any

infin

ite o

ccur

renc

ese

quen

ces.

Our

ver

sion

of C

TL

also

allo

ws

quer

ies

abou

ttr

an

sitio

ns a

nd b

ind

ing

ele

me

nts

.

•In

v(P

os(t

in A

rc))

chec

ks w

heth

er tr

ansi

tion

t is live.

Tim

ed C

P-n

ets

The

com

pute

r to

ols

for

CP

-net

s al

so s

uppo

rt s

tate

spac

e an

alys

is o

f time

d C

P-n

ets.

Col

oure

d P

etri

Net

s 88

Sta

te s

pace

s –

pro/

cont

raS

tate

spa

ces

are po

we

rfu

l and

easy

to u

se.

•T

he m

ain

draw

back

is th

e sta

te e

xplo

sio

n,

i.e.,

the

siz

e o

f th

e s

tate

sp

ace.

•T

he p

rese

nt v

ersi

on o

f our

tool

han

dles

gra

phs

with

250

,000

nod

es a

nd 1

,000

,000

arc

s. F

orm

any

syst

ems

this

is

no

t su

ffic

ien

t.

For

tuna

tely

, it i

s so

met

imes

pos

sibl

e to

con

stru

ctm

uch

mor

e co

mp

act

stat

e sp

aces

–

with

ou

t lo

osi

ng

info

rma

tion

.

•T

his

is d

one

by e

xplo

iting

the

inhe

rent

sym

me

trie

s of t

he m

odel

led

syst

em.

•W

e de

fine

two e

qu

iva

len

ce r

ela

tion

s (on

e fo

rm

arki

ngs

and

one

for

bind

ing

elem

ents

).

•T

he c

onde

nsed

sta

te s

pace

s ar

e of

ten

mu

chsm

alle

r (po

lyno

mia

l siz

e in

stea

d of

exp

onen

tial).

•T

he c

onde

nsed

sta

te s

pace

s co

ntai

n th

e sa

me

info

rma

tion

as th

e fu

ll st

ate

spac

es.

154

Col

oure

d P

etri

Net

s 89

Pla

ce in

varia

nts

anal

ysis

The

bas

ic id

ea is

sim

ilar

to th

e us

e of

in

varia

nts in

pro

gra

m v

erific

atio

n.

•A

pla

ce in

varia

nt is

an e

xpre

ssio

n whi

ch is

satis

fied

for

all r

each

able

mar

king

s.

•T

he e

xpre

ssio

n cou

nts

the

toke

ns o

f the

mar

king

– us

ing

a sp

ecifi

ed s

et o

f wei

ghts

.

We

first

co

nst

ruct

a se

t of p

lace

inva

riant

s.

The

n w

e ch

eck

whe

ther

they

are

fu

lfille

d.

•T

his

is d

one

by s

how

ing

that

eac

h oc

curr

ing

bind

ing

elem

ent re

spe

cts th

e in

varia

nts.

•T

he re

mo

ved s

et o

f tok

ens

mus

t be

iden

tical

toth

e a

dd

ed s

et o

f tok

ens

– w

hen

the

wei

ghts

are

take

n in

to a

ccou

nt.

Fin

ally

, we

use

the

plac

e in

varia

nts

to

pro

vebe

havi

oura

l pro

pert

ies

of th

e C

P-n

et.

•T

his

is d

one

by a

ma

the

ma

tica

l pro

of.

Col

oure

d P

etri

Net

s 90

Exa

mpl

e of

pla

ce in

varia

nts

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

(x,i)

e

if x=

q th

en 1

èel

se e

mpt

y

case

x o

f p

=>

2è

| q =

> 1`

e

2è

e

if x=

p th

en 1

èel

se e

mpt

y

e

case

x o

f p

=>

2è

| q =

> 1`

e

AP

3`(q

,0)

BP

2`(p

,0)

CP

DP

EPT

1

T2

T3

T4

T5

R E1è S E3è T E2è

INV

1: P

R1

INV

1: P

R1

2: Q

3: 2

*P+

2*Q

INV

1: P

R1

3: 2

*P+

2*Q

4: P

INV

1: P

R1

3: 2

*P+

2*Q

4: 2

*P+

Q

INV

2: ID

INV

3: ID

INV

4: ID

INV

AR

IAN

TS

1: 2

`p+

3`q

: U2:

1è

: E

3: 3

è :

E4:

2è

: E

INV

1: P

R1

2: Q

3: Q

155

Col

oure

d P

etri

Net

s 91

Pla

ce in

varia

nts

for

reso

urce

allo

catio

n sy

stem

To

spec

ify th

e w

eigh

ts w

e us

e thre

e f

un

ctio

ns:

•P

R 1 is

a p

roje

ctio

n fu

nctio

n: (

x,i)

-->

x.

•P

is a

n in

dic

ato

r fun

ctio

n: (

p,i)

-->

1è

; (q,

i) --

>Ø

.

•Q

is a

n in

dic

ato

r fun

ctio

n: (

p,i)

-->

Ø; (

q,i)

-->

1è.

•P

and

Q “

coun

ts”

the

num

ber

of p

and

q to

kens

.

PR 1

(M(A

)+M

(B)+

M(C

)+M

(D)+

M(E

)) =

2`p

+3`

q

M(R

) +

Q(M

(B)+

M(C

)) =

1è

M(S

) +

Q(M

(B))

+(2

*P+

2*Q

)(M

(C)+

M(D

)+M

(E))

= 3

è

M(T

) +

P(M

(D))

+ (

2*P

+Q

)(M

(E))

= 2

è

Col

oure

d P

etri

Net

s 92

A m

ore

read

able

ver

sion

of t

hepl

ace

inva

riant

s

PR 1

(A+

B+

C+

D+

E)

= 2

`p+

3`q

R +

Q(B

+C

) =

1è

S +

Q(B

) +

(2*

P+

2*Q

)(C

+D

+E

) =

3è

T +

P(D

) +

(2*

P+

Q)(

E)

= 2

è

The

pla

ce in

varia

nts

can

be u

sed

to

pro

vepr

oper

ties

of th

e re

sour

ce a

lloca

tion

syst

em, e

.g.,

that

it is

imp

oss

ible

to

re

ach

a d

ea

d m

ark

ing

.

156

Col

oure

d P

etri

Net

s 93

Too

l sup

port

for

plac

e in

varia

nts

Ch

eck

of p

lace

inva

ritan

s:

•T

he u

ser p

ropo

ses

a se

t of w

eigh

ts.

•T

he to

ol c

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itute

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ll pl

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s:

•T

his

is p

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ble,

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it is

a v

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lex t

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•M

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repr

esen

t the

res

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on a

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orm

, i.e

., a

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the

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ract

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Inte

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Col

oure

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etri

Net

s 94

How

to u

se p

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inva

riant

sIn

varia

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in o

rdin

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ram

min

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ua

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•In

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are

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urin

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157

Col

oure

d P

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Net

s 95

Pla

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varia

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– pr

o/co

ntra

Fro

m p

lace

inva

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is p

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to p

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beca

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com

bine

inva

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s of

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•T

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can

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com

pu

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the

data

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).

Col

oure

d P

etri

Net

s 96

Con

clus

ion

One

of t

he m

ain

reas

ons

for

the

succ

ess

of C

P-n

ets

is th

e fa

ct th

at w

e – si

mu

ltan

eo

usl

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ave

wor

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with

:

TO

OL

S •

edi

ting

• s

imul

atio

n •

veri

fica

tion

TH

EO

RY

• m

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s•

basi

c co

ncep

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veri

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met

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PR

AC

TIC

AL

USE

• s

peci

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• in

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• v

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icat

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•im

plem

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158

Col

oure

d P

etri

Net

s 97

Mor

e in

form

atio

n on

CP

-net

sT

he fo

llow

ing

WW

W p

ages

con

tain

a lo

t of i

n-fo

rmat

ion

abou

t CP

-net

s an

d th

eir

com

pute

r to

ols:

http

://w

ww

.dai

mi.a

u.dk

/CP

nets

/

A d

etai

led

intr

oduc

tion

to C

P-n

ets

can

be fo

und

inth

e fo

llow

ing

pape

rs/b

ooks

:L

.M.

Kri

ste

nse

n,

S.

Ch

rist

en

sen

an

d K

. Je

nse

n:

Th

e P

ract

itio

ne

r’s

Gu

ide

to

Co

lou

red

Pe

tri

Ne

ts.

In

t. J

ou

rna

l o

n S

oft

wa

re T

oo

ls f

or

Tec

hnol

ogy

Tra

nsfe

r, 2

(19

98),

Spr

inge

r V

erla

g, 9

5-19

1

K.

Jen

sen

: An

In

tro

du

ctio

n t

o t

he

Th

eo

retic

al

Asp

ect

s o

f C

olo

ure

dP

etr

i N

ets

. In

: J.

W.

de

Ba

kke

r, W

.-P

. d

e R

oe

ver,

G.

Roz

enbe

rg(e

ds.

): A

De

cad

e o

f C

on

curr

en

cy,

Le

ctu

re N

ote

s in

Co

mp

ute

rS

cien

ce v

ol. 8

03, S

prin

ger-

Ver

lag

1994

, 230

-272

.

K.

Jen

sen

: An

In

tro

du

ctio

n t

o t

he

Pra

ctic

al

Use

of

Co

lou

red

Pe

tri

Ne

ts. I

n:

W.

Re

isig

an

d G

. R

oze

nb

erg

(e

ds.

): L

ect

ure

s o

n P

etr

i Ne

tsII

: A

pp

lica

tion

s, L

ect

ure

No

tes

in C

om

pu

te S

cie

nce

Vo

l. 1

49

2,

Spr

inge

r-V

erla

g 19

98, 2

37-2

92.

K.

Jen

sen

: Co

lou

red

Pe

tri

Ne

ts.

Ba

sic

Co

nce

pts

, A

na

lysi

s M

eth

od

sa

nd

Pra

ctic

al

Use

. Mo

no

gra

ph

s in

Th

eo

retic

al

Co

mp

ute

r S

cie

nce

,S

prin

ger-

Ver

lag.

•V

ol. 1

: Bas

ic C

once

pts,

199

2, IS

BN

: 3-5

40-6

0943

-1.

•V

ol. 2

: Ana

lysi

s M

etho

ds, 1

994,

ISB

N: 3

-540

-582

76-2

.•

Vol

. 3: P

ract

ical

Use

, 199

7, IS

BN

: 3-5

40-6

2867

-3.

Col

oure

d P

etri

Net

s 98

Som

e of

the

mos

t im

port

ant p

aper

s on

hig

h-le

vel

nets

, the

ir ve

rific

atio

n m

etho

ds a

nd a

pplic

atio

nsha

ve b

een

repr

inte

d in

:K

. Je

nse

n,

G.

Ro

zen

be

rg (

ed

s.):

H

igh-

leve

l P

etr

i N

ets

. T

he

ory

an

dA

pp

lica

tion

. Spr

inge

r-V

erla

g, 1

991,

ISB

N: 3

-540

-541

25-X

.

A li

st o

f pap

ers

that

des

crib

e in

dust

rial u

se o

fC

P-n

ets

and

thei

r to

ols

can

be fo

und

on:

http

://w

ww

.dai

mi.a

u.dk

/CP

nets

/intr

o/ex

ampl

e_in

du.h

tml

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

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197

198

199

200

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203

204

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206

207

208

209

210

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212

213

214

215

216

ANINTRODUCTIONTO

GENERALIZEDSTOCHASTICPETRINETS

GianfrancoBalbo

DipartimentodiInformatica

UniversitadiTorino

1

Outline

•TIMEDPETRINETS

–Timedplaces,tokens,arcs,transitions

–Raceandpreselection

–Memory

–Singleandmultipleserversemantics

•STOCHASTICPETRINETS

–Theexponentialdistribution

–Markovchains

–IsomorphismbetweenSPNsandMCs

–Example

•GENERALIZEDSPNs

–Immediatetransitionsandpriority

–GSPNdefinition

–Extendedconflictsets

–IsomorphismbetweenGSPNsandMCs

–Performanceindices

–Example

•ANAPPLICATIONOFGSPNS

•CONCLUSIONS

2

217

Prerequisites

•ThebasicdefinitionsofPetrinettheory

–places,transitions,arcs,tokens

–marking

–enabling,firing,reachability

–(enablingdegree)

–conflict,confusion

–(invariants)

•Someelementarynotionsofprobabilitytheory

–randomvariable

–stochasticprocess

–pdf,PDF

–statespace

–averages

–sojourntimes

–(ergodicity)

–(Little’sformula)

Itemsinparenthesesareoptional.

3

TIMEDPETRINETS

4

218

Timingspecifications

TimeisintroducedinPetrinetstomodeltheinteraction

amongseveralactivitiesconsideringtheirstartingand

completiontimes

Theintroductionoftimespecificationscorrespondsto

aninterpretationofthemodelbymeansof

•observationoftheautonomous(untimed)model

•definitionofanon-autonomousmodel

Timespecificationsshouldprovide

•consistencyamongautonomousandnon-autonomous

models

•non-determinismreductiononthebasisoftimecon-

siderations

•supportforthecomputationofperformanceindices

5

Timedplaces

Severalapproachesarepossiblefortheintroductionof

temporalspecificationsinPNmodels:

•timemaybeassociatedwithplaces(TPPN):

–tokensgeneratedinanoutputplacebecomeavail-

abletofireatransitiononlyafteradelayhas

elapsed;thedelayisanattributeoftheplace

6

219

Timedtokens

•timemaybeassociatedwithtokens:

–tokenscarryatimestampthatindicateswhen

theyareavailabletofireatransition;thistime

stampcanbeincrementedateachtransitionfir-

ing.

7

Timedarcs

•timemaybeassociatedwitharcs:

–atravellingdelayisassociatedwitheacharc;to-

kensareavailableforfiringonlywhentheyreach

atransition

8

220

Timedtransitions

•timemaybeassociatedwithtransitions(TTPN);

transitionsrepresentactivities

–activitystartcorrespondstotransitionenabling,

–activityendcorrespondstotransitionfiring

Differentfiringpoliciesmaybeassumed:

–three-phasefiring

1.tokensareconsumedfrominputplaceswhen

thetransitionisenabled

2.thedelayelapses

3.tokensaregeneratedinoutputplaces

–atomicfiring

tokensremainininputplacesforthetransition

delay;theyareconsumedfrominputplacesand

generatedinoutputplaceswhenthetransition

fires

9

Atomicfiring

WeshallconsiderTTPNwithatomicfiring.

TTPNwithatomicfiringcanpreservethebasicbe-

haviouroftheunderlyinguntimedmodel.

ItisthuspossibletoqualitativelystudyTTPNwith

atomicfiringexploitingthetheorydevelopedforun-

timed(autonomous)PN(reachabilityset,invariants,etc.).

Timingspecificationsmayaffectthequalitativebehaviour

ofthePNwhentheydescribe

const

antand

inte

rvalfir-

ingdelays.

10

221

Internaltimer

Wecanexplainthebehaviourofonetimedtransition

withatomicfiringbyassumingthatitincorporatesa

timer.

•Whenthetransitionisenabled,itstimerissettothe

currentdelayvalue

•Then,thetimerisdecrementedatconstantspeed,

untilitreachesthevaluezero

•Atthispointthetransitionfires

11

Conflicts

Whenmorethanonetimedtransitionwithatomicfiring

isenabled,thebehaviourissimilar,butaproblemarises:

Whic

hone

ofth

eenabled

transi

tions

isgoin

gto

fire

?

12

222

Selectionrules

Twoalternativeselectionrules:

•preselection:

theenabledtransitionthatwillfireischosenwhen

themarkingisentered,accordingtosomemetric

(priority,probability,...)

•race:

theenabledtransitionthatwillfireistheonewhose

firingdelayisminimum

13

Memorypolicies

Whenatimedtransitionisdisabledbyaconflictingtran-

sition,aproblemarises:

How

isth

etr

ansi

tion

tim

er

setwhen

the

transi

tion

will

again

beco

me

enabled?

How

does

the

transi

tion

kee

pmemory

of

its

past

en-

abling

tim

e?

14

223

Basicmechanisms

Twobasicmechanismscanbedefined:

•Continue:

thetimerassociatedwiththetransitionholdsthe

presentvalueandwillco

ntinuelateronthecount-

down

•Restart:

thetimerassociatedwiththetransitionis

rest

arte

d,

i.e.,itspresentvalueisdiscardedandanewvalue

willbegeneratedwhenneeded

15

Transitionmemorypolicies

Fromthetwobasicmechanismsitispossibletocon-

structseveraltransitionmemorypolicies;theusualones

are: •Resampling:

–Ateachandeverytransitionfiring,thetimersof

alltimedtransitionsinthetimedPNsystemare

discarded(restartmechanism).

–Nomemoryofthepastisrecorded.

–Afterdiscardingalltimers,newvaluesofthetimers

aresetforthetransitionsthatareenabledinthe

newmarking.

16

224

•Enablingmemory:

–Ateachtransitionfiring,thetimersofalltimed

transitionsthatbecomedisabledarerestarted,

whereasthetimersofalltimedtransitionsthat

remainenabledholdtheirpresentvalue(continue

mechanism).

–Thememoryofthepastisrecordedwithan

en-

abling

mem

ory

vari

ableassociatedwitheachtran-

sition.

–Theenablingmemoryvariableaccountsforthe

workperformedbytheactivityassociatedwith

thetransitionsincethelastinstantoftimewhen

itstimerwasset.

Theenablingmemoryvariablemeasurestheen-

ablingtimeofthetransitionsincethelastinstant

oftimeitbecameenabled.

17

•Agememory:

–Ateachtransitionfiring,thetimersofalltimed

transitionsholdtheirpresentvalues(continuemech-

anism).

–Thememoryofthepastisrecordedwithan

age

mem

ory

vari

ableassociatedwitheachtimedtran-

sition.

Theagememoryvariableaccountsforthework

performedbytheactivityassociatedwiththetran-

sitionsincethetimeofitslastfiring.

–Theagememoryvariablemeasuresthe

cum

ula

-

tiveenablingtimeofthetransitionsincethelast

instantoftimewhenitfired.

18

225

Transitionenabling

The

enabling

deg

reeofatransitionisthenumberof

timesthetransitioncouldfireinthegivenmarkingbe-

forebecomingdisabled.

Whentheenablingdegreeofatransitionis>1,attention

mustbepaidtothetimingsemantics.

19

Serversemantics

Threecasesarecommon:

•Single-serversemantics

•Infinite-serversemantics

•Multiple-serversemantics

20

226

Sin

gle-s

erv

er

sem

antics:

Afiringdelayissetwhenthetransitionisfirstenabled,

andnewdelaysaregeneratedupontransitionfiringif

thetransitionisstillenabledinthenewmarking.

Enablingsetsoftokensareprocessed

seria

llyandthe

temporalspecificationassociatedwiththetransitionis

independentoftheenablingdegree;

21

Infinite-s

erv

er

sem

antics:

Everyenablingsetoftokensisprocessedassoonasit

formsintheinputplacesofthetimedtransition.

Itscorrespondingfiringdelayisgeneratedatthistime,

andthetimersassociatedwithalltheseenablingsets

rundowntozeroconcurrently.

Multipleenablingsetsoftokensarethusprocessed

in

para

llel.

Theoveralltemporalspecificationsoftransitionswith

thissemanticsdependdirectlyontheirenablingdegrees

22

227

Multip

le-s

erv

er

sem

antics:

Enablingsetsoftokensareprocessedassoonasthey

formintheinputplacesofthetransitionuptoamaxi-

mumdegreeofparallelism(sayK).

Forlargervaluesoftheenablingdegree,thetimersasso-

ciatedwithnewenablingsetsoftokensaresetonlywhen

thenumberofconcurrentlyrunningtimersdecreasesbe-

lowthevalueK.

Theoveralltemporalspecificationsoftransitionswith

thissemanticsdependdirectlyontheirenablingdegrees

uptoathresholdvalueK

23

Exampleofserversemantics

Consideratimedtransitionwithenablingdegreeequal

to3.

Thethreeenablingsareassociatedwithfiringdelays

equalto3,2,and4timeunits.

t3

24

0t

32

0

t0

59

3

6

24

228

Queueingpolicies

Uponfiringofatransition,inputtokensareremovedat

random

Specificqueueingpoliciesmustbeexplicitlyrepresented

atmodellevel

25

Firingandselectionrules

WeconsiderTTPNwithatomicfiringandraceselection

rule.

TransitionswithinoneTTPNcanuse

•Resampling

•Enablingmemory

•Agememory

and •Single-serversemantics

•Infinite-serversemantics

•Multiple-serversemantics

inanycombination

26

229

STOCHASTICPETRINETS

27

Probabilisticinterpretation

TimedTransitionPNwithatomicfiringinwhichall

transitiondelaysare

random

varia

ble

swithnegativeex-

ponentialdistributionsarecalled

Sto

chast

icPN

(SPN

).

ThedynamicbehaviourofaSPNisdescribedthrough

ast

ochast

icpro

cess.

28

230

Definitions

Ara

ndom

varia

bleisarealfunctiondefinedoveraprob-

abilityspace.

Sto

chast

icpro

cess

esaremathematicalmodelsusefulfor

thedescriptionofphenomenaofaprobabilisticnatureas

afunctionofaparameterthatusuallyhasthemeaning

oftime.

Astochasticprocess

{X(t),t∈T}isafamilyofrandom

variablesdefinedoverthesameprobabilityspace,in-

dexedbytheparametertandtakingvaluesinthestate

spaceS.

29

Stochasticprocesses

Asa

mple

path(orrealization)ofastochasticprocessisa

functionoftime.

Con

tinuo

us ti

me

Dis

cret

e tim

e

X(t

)X

n

tn

05

1015

20

30

231

TheprobabilisticdescriptionofarandomvariableXis

givenbyits

pro

babi

lity

den

sity

funct

ion(pdf)

f X(x)=d dxP{X

≤x}

−∞<x<

∞

Theprobabilisticdescriptionofarandomprocessisgiven

bythejointpdfofanysetofrandomvariablesextracted

fromtheprocess.

P{X

(t1)≤x

1,X(t

2)≤x

2,...,X(tn)≤xn}

Inthegeneralcasethecompleteprobabilisticdescription

ofarandomprocessisnotfeasible.

MA

RK

OV

IAN

pro

cess

esareonespecialclassofstochas-

ticprocessesforwhichtheprobabilisticdescriptionis

simplerandofparticularrelevance.

31

AprocessthatsatisfiestheMarkovproperty:

P{X

(t)≤x|X(tn)=xn,X(tn−1)=xn−1,···

X(t

0)=x

0}=

P{X

(t)≤x|X(tn)=xn}

witht>t n>t n

−1>

···>t 0iscalleda

Mark

ovi

an

pro

cess.

Ifthestatespaceisdenumerable,theprocessisaM

arkov

chain.

Iftheparametertiscontinuous,theprocessisaco

ntinuous-

tim

eMarkovchain(CTMC). 32

232

Acontinuous-timeMarkovchain(CTMC)isastochastic

processwhere

•sojourntimesinstatesareexponentiallydistributed

randomvariables

•thefutureevolutiondependsonlyonthepresent

state,notonthepasthistory

33

Exponentialdistributions

Theexponentialpdf

f X(x)=λe−λx

(x≥0)

istheonlycontinuouspdfforwhichthememoryless

propertyholds:

P{X

>x+α|X>α}=

P{X

>x}

X

x0

a

�-

x

e�

nor

mal

izat

ion

34

233

Theexponentialpdf

f X(x)=λe−λx

(x≥0)

isdefinedonlybyits

rateλ,whichistheinverseofits

averagevalue:

E[X]=1 λ

35

GiventworandomvariablesXandYwithexponential

pdf

f X(x)=λe−λx

(x≥0)

f Y(y)=µe−µy

(y≥0)

thenewrandomvariableZ=min(X,Y)alsohasanex-

ponentialpdf f Z(z)=(λ+µ)e

−(λ+µ)z

(z≥0)

Infact,

FZ(z)=

1−Pr{Z>z}

=1

−Pr{X>z,Y>z}

=1

−e−λze−µz=1

−e−

(λ+µ)z

(z≥0)

36

234

Markovchains

TheresidualsojourntimeinastateofaMarkovchain

isarandomvariablewiththesamedistributionasthe

wholesojourntime.

37

ACTMCcanbedescribedthrougha

state

transi

tion

rate

dia

gram,orequivalentlywitha

state

transi

tion

rate

matr

ix,

alsocalled

infinites

imalge

ner

ato

r,denotedby

Q.

Q=

−λ

λ

µ−µ

38

235

ThesolutionofaCTMCattimetistheprobability

distributionoverthesetofstates:

π(t)=(π

1(t),π

2(t),π

3(t),···)

with

πi(t)=P{X

(t)=i}

Itcanbeproventhat dπ(τ)

dτ

=π(τ)Q

whosesolutioncanbeformallywrittenas

π(t)=

π(0)H

(t)

with

H(t)=eQ

t

Thisisaveryelegantsolutionthatishoweverusually

veryexpensivetocomputesincethematrixexponenti-

ationisdefinedbythefollowinginfinitesum

eQτ=

∞ ∑ k=

0

(Qτ)k

k!

39

ThesolutionofaCTMCatsteady-stateistheprobabil-

itydistributionoverthesetofstates.

Thesteady-statedistributionexistsfor

ergo

dicCTMCs.

Thesteady-statedistribution

π=(π

1,π

2,π

3,···)

with

πi=lim

t→∞P{X

(t)=i}

iscomputedasthesolutionofthelinearsystemofequa-

tions

πQ=0

withthenormalizingcondition

∑ iπi=1

40

236

DefinitionofstochasticPetrinets

Formally,anSPNisdefinedthroughan8-tuple:

SPN=(P,T,I(.),O(.),H(.),W(.),M

0)

where

•PN

=(P,T,I(.),O(.),H(.),M

0)isthemarkedPN

underlyingtheSPN

•W(.)isafunctiondefinedonthesetoftransitions

thatassociatesaratewitheachtransition.Thisrate

istheinverseoftheaveragefiringtimeofthetran-

sition

41

SPNscanbeprovedtobeisomorphictoCTMCs:the

reachabilitygraphoftheSPNcorrespondstothestate

transitionratediagramoftheMC.

Thiscanbeeasilyseeninthecaseofsimplesubclassesof

Petrinetssuchas:

Fin

ite

Sta

teM

achin

esand

Mark

ed

Gra

phs

42

237

SPNswithoutchoicesand

synchronizations

•ThenethasthestructureofbothaFiniteStateMa-

chine(notransitionhasmorethanoneinputandone

outputplace)andofaMarkedGraph(noplacehas

morethanoneinputandoneoutputtransition);

•theinitialmarkingcontainsonlyonetoken

Eachplaceofthenetunivocallyidentifiesastateofthe

net.

Eachplaceofthenetmapsintoastateofthecorre-

spondingprobabilisticmodel.

Thetimespentbythetokenineachofitsplacesis

completelydeterminedbythecharacteristicsoftheonly

transitionthatcanwithdrawitfromthatplace.

43

Theprobabilisticmodelthatrepresentsthebehaviour

ofthenet(M

ark

ing

Pro

cess)isaCTMC

p 1p 2

p 3T

3T

2T

1

γµ

λ

1 0

00

1 0

0 0

1

λµ

γ 44

238

SPNswithchoices

•ThenethasthestructureofaFiniteStateMachine

(notransitionhasmorethanoneinputandoneout-

putplace);

•theinitialmarkingcontainsonlyonetoken

Conflictsarisewhenseveraltransitionsshareacommon

inputplace

Aracestartsamongthesimultaneouslyenabledtransi-

tions.

Theraceiswonbyoneofthetransitionsandtheway

ofdealingwiththepartiallycompletedactivitiesofthe

transitionsthatwereinterruptedbecomesanissue(in

general).

45

Whenthefiringtimesofthetransitionsofthenethave

negativeexponentialdistributions,theirmemorylessprop-

ertymakesthedistinctionamong

•resampling

•enablingmemory

•agememory

irre

levantandtheCTMCcorrespondingtotheSPNis

obtainedfromthenetinastraightforwardmanner.

p 2

p 1p 3 p 4

T5

T3

T4

T1

T2

σγ

δ

λ

µ

1 0

0 0

0 1

0 0

0 0

1 0

0 0

0 1

λµ γ

δ σ 46

239

Morecomplexsituationsarisealreadywhenseveralto-

kensareallowedintheinitialmarkingsofthesesimple

models.

•Servicesemanticsadoptedwhentheinputplaceofa

transitioncontainsseveraltokens;

•Queueingpolicyassumedwithrespecttothetokens

residingintheinputplaceofatransition.

47

Asimpleexample

p 2p 1

T1

T2

λµ

IS(onbothtransitions)

0 2

1 1

2 0

λ

µµ 2

λ 2

SS(onbothtransitions)

0 2

1 1

2 0

λ

µλ

µ 48

240

Queuein

gpolicy

Itispossibletoshowthatwhenthefiringtimesareex-

ponentiallydistributedandtheperformancefiguresof

interestareonlyrelatedtothemomentsofthenumber

oftokensintheinputplaceofatransitionmanyqueue-

ingpoliciesyieldthesameresultsandthusthe

random

ord

er(thatisthemostnaturalinthePetrinetcontext)

canbeassumed.

49

Ingeneral,theCTMCassociatedwithagivenSPNsys-

temisobtainedbyapplyingthefollowingsimplerules:

1.TheCTMCstatespaceS=

{si}correspondstothe

reachabilitysetRS(m

0)ofthePNassociatedwiththe

SPN(m

i↔s i).

2.Thetransitionratefromstates i(correspondingto

marking

mi)tostates j(m

j)isobtainedasthesum

ofthefiringratesofthetransitionsthatareenabled

inmiandwhosefiringsgeneratemarking

mj.

Assumingthatallthetransitionsofthenetoperate

withasingle-serversemanticsandmarking-independent

speeds,anddenotingwith

•Qthe

infinitesi

malgenera

tor,

•wkthefiringrateofTk,

•e

j(m

i)=

{h:Th∈

e(m

i)∧

mi[Th〉m

j}thesetoftransi-

tionsthatbringthenetfrom

mito

mj,

thecomponentsofQare:

q ij=

∑T

k∈e

j(m

i)wk

−qi

i�=j

i=j

where

q i=

∑

Tk∈e

(mi)wk

50

241

Performanceindices

Thesteady-statedistribution

πisthebasisforaquanti-

tativeevaluationofthebehaviouroftheSPNexpressed

intermsofperformanceindices.

Theseresultscanbecomputedusingaunifyingapproach

inwhichproperindexfunctions(alsocalled

reward

func-

tions)aredefinedoverthemarkingsoftheSPNandan

averagerewardisderivedusingthesteady-stateproba-

bilitydistributionoftheSPN.

Assumingthatr(

m)representsoneofsuchrewardfunc-

tions,theaveragerewardcanbecomputedusingthe

followingweightedsum:

R=

∑

mi∈RS

(m0)r(

mi)πi

51

Pro

bability

ofa

particula

rco

nditio

nΥ(m

)ofth

eSPN

.

Definethefollowingrewardfunction:

r(m)=

1 0

Υ(m

)=true

otherwise

Thedesiredprobabilityiscomputedusingthefollowing

expression:

P{Υ

}=

∑

mi∈RS

(m0)r(

mi)πi=

∑

mi∈Aπi

whereA

={m

i∈RS(m

0):Υ(m

i)=true}.

52

242

Expec

ted

valu

eofth

enum

berofto

kensin

agiv

en

pla

ce.

Inthiscasetherewardfunctionis:

r(m)=niff

m(pj)=n

Theexpectedvalueofthenumberoftokensinp jisgiven

by: e[m(pj)]=

∑

mi∈RS

(m0)r(

mi)πi==

∑ n>

0[nP{A(j,n)}]

whereA(j,n)=

{mi∈RS(m

0):

mi(p j)=n}andthe

sumisobviouslylimitedtovaluesofn≤k;

iftheplaceisk-bounded.

53

Mea

nnum

ber

of

firi

ngs

per

unit

of

tim

eof

agiv

en

transi

tion.

Atransitionmayfireonlywhenitisenabled,thusthe

rewardfunctionassumesthefollowingform:

r(m)=

wj 0

Tj∈

e(m

)

otherwise

ThemeannumberoffiringsofTjperunitoftimeisthen

givenby: f j=

∑

mi∈RS

(m0)r(

mi)πi==

∑

Mi∈A

j

wjπi

whereAj=

{mi∈RS(m

0):Tj∈

e(m

i)}.

54

243

Theaveragesteady-statedelayspentintraversingasub-

networkcanbecomputedfromLittle’sformla

E[T]=E[N]

E[S]

whereE[N]istheaveragenumberof(equivalent)tokens

inthesubnet,andE[S]istheaverageinputrateintothe

subnet.

Delaydistributionsareingeneraldifficulttocompute.

55

Example

TheSPNdescriptionofasimpleparallelsystem

P1

P2

Tnewdata

Tpar1

P3

P5

Tstart

Tpar2

P4

P6

Tcheck

P9

Tsyn

TKO

P7

TI/O

P8

TOK

56

244

P1 P2

P3+

P4

P5+

P6

P7

P9P8

P4+

P5P3

+P6

W(T

new

data

)

star

tW

(T)

chec

kW

(T)

syn

W(T

)

par1

W(T

)pa

r2W

(T)

I/O

W(T

)

µ 2µ 1 K

Oλ

OK

λ

StatespaceforM(P1)=1

57

Com

puta

tion

ofµ

1,µ

2:

•TotalrateoutofP3+P4is:

W(Tpar1)+W(Tpar2)

•WithwhatprobabilityTpar1isthefirsttofire?

W(Tpar1)

W(Tpar1)+W(Tpar2)

•Therefore:

µ1=(W

(Tpar1)+W(Tpar2))

W(Tpar1)

W(Tpar1)+W(Tpar2)

=W(Tpar1)

58

245

Com

puta

tion

ofλOK,λKO:

samecomputationasbefore:

λOK=W(TOK)

λKO=W(TKO)

but... whatisthemeaningofW(TOK)andW(TKO)?

→checkactivity:0.0001

→rateof10,000

→probabilityofOK/KOis99%vs.1%

→W(TOK)=9,900andW(TKO)=100

59

Para

mete

rsp

ecifi

cations:

transitionratevalue

semantics

Tnewdata

λ1

infinite-server

Tstart

τ1000single-server

Tpar1

µ1

10

single-server

Tpar2

µ2

5single-server

Tsyn

σ2500single-server

TOK

α9900single-server

TKO

β100

single-server

TI/O

ν25

single-server

Tcheck

θ0.5

single-server

Theconsistencycheckoperationhasanaverageduration

0.0001timeunits,andresultsinasuccess99%ofthe

times,andinafailure1%ofthetimes.

60

246

Perf

orm

ance

indic

es:

•ThroughputoftransitionTI/O:

–1.504success/timeunits

•Averagenumberofitemsundertest:

–0.031

•Averageproductiontime:

–0.33timeunits

61

GENERALIZEDSTOCHASTIC

PETRINETS

62

247

TwoclassesoftransitionsexistinGSPNs:

•tim

edtransitions,whosedelaysareexponentiallydis-

tributedrandomvariables(likeinSPNs)

•im

med

iatetransitions,whosedelaysaredeterminis-

ticallyzero

Immediatetransitionshavebeenintroducedinthemodel

•toaccountforinstantaneousactions(e.g.choiceamong

classesofclients);

•toimplementspecificmodellingfeatures(e.g.to

emptyaplace);

•toaccountfortimescaledifferences(e.g.busarbi-

trationandI/Oaccesses).

63

Immediatetransitionshavepriorityovertimedtransi-

tions.

Severalprioritylevelsforimmediatetransitionscanbe

defined.Immediatetransitionsatprioritylevelnare

calledn-immediate.

TheautonomousmodelassociatedwithaGSPNisa

Petr

inetwith

pri

ori

ties.

Atransitiontissaidtohave

conce

ssio

ninmarkingM

iffM

≥I(t)

∧M<H(t).

Atransitiont jisdefinedtobeenabledinmarkijgMiff

ithasconcessioninMand

∀tk∈Tthathaveconcession

inM,πj≥πk.

64

248

Effectsinducedbythepresenceofpri-

orities

Σ⇐⇒

Σπ

•Properties

→safety(invariant):mustholdinallstates

→eventuality(progress):mustholdinsomestate

•RS(Σ)⊇RS(Σπ)

→safetypropertiesaremaintained(absenceofdead-

locks,boundedness,mutualexclusion,...)

→eventualitypropertiesarenotnecessarilymain-

tained(reachability,liveness,...)

65

Effectsinducedbythepresenceofpri-

orities

Σ⇐⇒

Σπ

•Reachability

→M

∈RS(Σ)�⇒M

∈RS(Σπ)

but

→M

∈RS(Σπ)⇒M

∈RS(Σ)

•Boundedness

→Σbounded⇒Σπbounded

but

→Σnotbounded�⇒Σπnotbounded

t1p1

t2

t3p2

t4

•Liveness-homestates

→prioritycanintroduceorremoveliveness

66

249

Somefinepoints:

1.Needforpriority

p1

2.Irrelevanceofdistinctionbetweenresampling-en-

abling-ageduetothememorylesspropertyofex-

ponentialdistributions

3.Impossibilityoftwotimerstoexpireatthesametime

probabilityofextractingaspecificsam-

plexisequaltozero

67

Markingsthatenabletimedtransitionsonlyaresaidto

be

tangib

le,whereasmarkingsthatenablen-immediate

transitionsaresaidtobe

vanishin

g.

p1

p2

p3p4

T1

T2

T3

t1t2

αβ

µ

λ 1λ 2

p2p1

p3p4

T3

T1

t1t2

T2

p1

p3p4

λ 1λ 2

µα+

βαµ

α+ββ

68

250

Definitionof

generalizedstochasticPetrinets

Formally,aGSPNisan8-tuple:

GSPN=(P,T,Π(.),I(.),O(.),H(.),W(.),M

0)

where

•PNπ=

(P,T,Π(.),I(.),O(.),H(.),M

0)isthemarked

PNwithpriorityunderlyingtheGSPN

•W(.)isafunctiondefinedonthesetoftransitions

Thesubnetsformedbyn-immediatetransitionsmustbe

confusion-free.

69

ThefunctionW(.)allowsthedefinitionofthestochastic

componentofaGSPNmodel.Inparticular,itmaps

transitionsintorealpositivenumbers.

ThequantityW(tk)=wkiscalled

•the“rate”oftransitiont kift kistimed

•the“weight”oftransitiont kift kisn-immediate

RatesareusedlikeinSPNs.

Weightsareusedfortheprobabilisticresolutionofcon-

flictsofimmediatetransitions.

70

251

Whenatangiblemarkingisentered,thetimedtransi-

tionsthatbecomeenabledforthefirsttimesincetheir

lastfiring,sampleafiringdelayinstanceandsettheir

timertothesampledvalue.

Then,alltimersoftheenabled(timed)transitionsare

decrementedatequalspeed,untiloneofthemreaches

thevaluezero.

Atthispointthetransitionwhosetimerreachedzero

fires.

Allthetransitionsthatdidnotfirekeeptheirtimerread-

ings,andtheirtimerswillbeagaindecrementedinthe

nextmarkinginwhichthetransitionisenabled.

(Enabling

mem

ory

was

use

din

the

desc

ription,but

is

irre

levant)

71

Whenavanishingmarkingisentered,theweightsofthe

enabledn-immediatetransitionsareusedtoprobabilis-

ticallyselectthe(n-immediate)transition(s)tofire.The

timespentinanyvanishingmarkingisdeterministically

equaltozero.

72

252

p2+

p4p3

p1+

p4

W(t

1)W

(t1)

+W(t

2)

p1

p2p3

p4

p5

p6

t1t2

t3

W(t

2)W

(t1)

+W(t

2)

W(t

1)=10

W(t

2)=20

W(t

3)=44

note:

•Theweightoft 2withrespecttot 1isal-

waysthesame,regardlessofwhethert 3

isenabledornot.

•Extensionstomarkingdependentrates

havebeendefined. 73

Thedefinitionofweightsrequirestheidentificationof

thesetsofimmediatetransitionsthatcanbesimultane-

ouslyenabledinconflict.

SuchsetsoftransitionsarecalledExtendedConflictSets

(ECSs).

WhenalltheECSsinaGSPNareknown,theassocia-

tionofweightstotransitionsiseasy,providedthatno

confu

sionexists.

Thestructuralandbehaviouralanalysisofthemarked

PNwithpriorityunderlyingtheGSPNallowsthequal-

itativestudyoftheGSPNbehaviour,andinparticular

theidentificationof

•ECS

•confusion

74

253

Confusiondestroysthelocalityofconflicts

p 0p 1

p 3 p 4p 2

p a

p 0p 1

p 3 p 4p 2

(b)

(a)

t 0

t 2t 1

t 2t 1

t at 0

w0

αβ

αβ

wa

w0

75

Confusionproducesdifferenttransitionprobabilities

p0+

p2

p0+

p4p1

+p2

p1+

p4p2

+p3

t2t0

t2t0

t1

1

αα+

ω0

ω0

α+ω0

α α+β

β α+β

p0+

p2

t2t0

t2t1

αα+

ω0

ω0

α+ω0

α α+β

β α+βp1

+p2

p0+

p4pa

+p2

p2+

p3p1

+p4

pa+

p4

1 1

t0 ta

t2ta

αα+

ω0

ω0

α+ω0

76

254

Inthecaseofvanishingmarkings,theweightsoftheen-

abledn-immediatetransitionscanbeusedtodetermine

whichonewillactuallyfireinamarkingMthatenables

morethanoneconflictingn-immediatetransitions.

WhenseveraltransitionsbelongingtothesameECSare

theonlyonesenabledinagivenmarking,oneofthem,

saytransitiont i,isselectedasacandidatetofirewith

probability:

P{t i|M

}=

wi

WI(M

)

whereWI(M

)istheweightofECS(ti)inmarkingM,and

isdefinedasfollows:

WI(M

)=

∑

k:t k∈ECS

(ti)∩E

(M)wk

77

ItmayhoweverhappenthatseveralECSscomprising

transitionsofthesameprioritylevelaresimultaneously

enabledinavanishingmarking.

Thecharacteristicofthesubnetsofn-immediatetransi-

tionsofbeingconfusion-freeguaranteesthatthewayin

whichthischoiceisperformedis

irre

levantwithrespect

totheresultingstochasticmodel.

78

255

GSPNscanbeprovedtobeisomorphictoSemi-Markov

processes.

TheanalysisofaGSPNcanbeperformedbystudying

aCTMC.

ThestatetransitionratediagramoftheMCcorresponds

tothe

tangib

lereachabilitygraphoftheGSPN.

Thememorylesspropertyoftheexponentialdistribution

makesthedistinctionamong

•resampling

•enablingmemory

•agememory

irrelevant.

79

Thesojourntimeinatangiblemarkingisexponentially

distributedwithaparameterthatisthesumoftherates

ofallenabledtimedtransitions,sothattheaveragetime

spentinmarkingMisgivenby:

E[SJ(M

)]=

∑

t∈E

(M)W(t) −1

80

256

NumericalsolutionofGSPNmodels

AnembeddedMarkovchain(EMC)canberecognized

disregardingtheconceptoftimeandfocusingtheatten-

tiononthesetofstatesofthesemi-Markovprocess.

ThespecificationsofaGSPNsystemaresufficientfor

thecomputationofthetransitionprobabilitiesofsucha

chain.

Severaltechniqueshavebeendevisedforrestrictingthe

computationtoreducedmodelsaccountingforthetan-

giblemarkingsonly.

81

Threedifferentapproachescanbeused:

•IdentifyareducedEmbeddedMarkovChaindefined

overthesetoftangiblemarkingsonly;

•Computethetransitionprobabilitiesamongtangible

markingsdirectlybyapplying(on-the-fly)adepth-

firstalgorithmthatexploresallcompletevanishing

pathsemanatingfromeachtangiblestate.

Themethodassumesthatnoloopsamongvanish-

ingstatesexistandmemorysavingistraded-offwith

(possible)repeatedcomputations;

•ReducetheGSPNtoanequivalentSPNobtained

byfusingimmediatetransitionswithprecedingtimed

transitionsusinganalgorithmthatinthesimplecases

producesthefollowingreductions

p 1p 3

p 2p ap a

p 3p 2

p 1

p a

T1

T2

T3

Ta

t 3t 2

t 1

82

257

Computationalconsiderations

Themathematicallyelegantsolutionofthemodelus-

ingtheREMCsuffersinpracticeofthedifficultiesde-

rivingfromthesizeoftheCTMCandfromtime-scale

differencesthatmayexistamongthefiringratesofthe

transitionsofamodel.

Approachesthatcanbeusedtoovercomethesedifficul-

tiesarethefollowing:

•Transientsolution

–Uniformizationmethod

•Steadystatesolution

–Timescaledecomposition

–Tensoralgebrasandcompositionality

–Symmetriesandexactlumping

•Simulation

83

Performanceindices

Fromthesteady-stateprobabilitydistributionofmark-

ingsitispossibletoobtainseveralperformanceindices

thatarethebasisforaquantitativeevaluationofthe

behaviouroftheGSPN.

AsinthecaseofSPNs,theseresultscanbecomputed

usingtheunifyingapproachbasedonthedefinitionof

rewardfunctions.

84

258

Tools

Theapplicabilityofthe[(G)S]PNapproachtoanything

butthesmallesttoyexamplesrestsontheavailabilityof

efficienttoolsforthe

•modelconstruction(top-d

own,bo

ttom

-up,co

mposi

-

tionality)

•modeldebugging(s

tructu

ralanaly

sis)

•definitionofperformanceindices

•modelsolution(a

naly

sis

and/or

sim

ula

tion)

•computationofaggregateresults

•displayofresults

Goodsoftwaretoolsareamust.

Theuser-friendlinessandthegraphicalcapabilitiesof

thetoolareofparamountimportance.

85

Example

TheGSPNdescriptionofasimpleparallelsystem

P1

P2

Tnewdata

Tpar1

P3

P5

t start

Tpar2

P4

P6

Tcheck

P9

t syn

t KO

P7

TI/O

P8

t OK

86

259

transitionratevalue

semantics

Tnewdata

λ1

infinite-server

Tpar1

µ1

10

single-server

Tpar2

µ2

5single-server

TI/O

ν25

single-server

Tcheck

θ0.5

single-server

transitionweightpriorityECS

t start

11

1

t syn

11

2

t OK

99

13

t KO

11

3

Theconsistencycheckoperationresultsinasuccess99%

ofthetimes,andinafailure1%ofthetimes.

87

TheGSPNmodelgenerates

•20tangiblemarkings

•18vanishingmarkings

Asanexample,withthenumericalvalueschosenforthe

modelparameters,theprobabilityofatleastoneprocess

waitingforsynchronizationiscomputedtobe0.238.

88

260

Acasestudy

akanbansystem

CA

RD

CE

LL

CE

LL

i -

1 i

+ 1

PA

RT

PA

RT

CA

RD

MA

CH

INE

INP

UT

BU

FF

ER

BU

FF

ER

OU

TP

UT

JOIN BO

AR

DB

UL

LE

TIN

CE

LL

i

Akanbancellandthepartsandcardsthatflowintoand

outofit

89

Thebasicmodel

idleM

i

OB

ibusyM

iIB

i

BB

i

outM

iexitCell i

inM

ienterCell i

Ki

GSPNmodelofasingleKanbancell

90

261

n-cellsequentialKanban

BB

n−1

IB

n

BB

3

OB

2

IB

2

OB

1

IB

1

idleM

1

BB

1

busyM

1

idleM

2busyM

2

BB

2

BB

n

busyM

nidleM

n

OB

n

outM

n

outM

1

outM

2

exitCell n

inM

1

enterCell 1

enterCell 2

inM

2

enterCell 3

inM

n

enterCell n

K1

K2

K3

Kn−1

Kn

91

Qualitativeanalysis

An-cellKanbanmodelhas2nminimalP-semiflows,whose

associatedP-invariantsare:

∀i,1

≤i≤n:

M(BBi)

+M(IBi)

+M(busyMi)+M(OBi)=Ki

M(idleMi)+M(busyMi)=

1

Itfollowsthat:

•ThenumberofpartsincelliisatmostKi,thenum-

berofcardsinthecell;

•Eachmachinecanprocessonlyonepartatatime;

•PlacesidleMiandbusyMiaremutuallyexclusive.

92

262

AlltransitionsoftheGSPNmodelarecoveredbya

sin-

gleminimalT-semiflow:itrepresentsthedeterministic

flowoftheuniquetypeofpartsprocessedbythesystem.

Thenetbehaviourisdeterministic:nostructuralcon-

flictsexist,henceneithereffectiveconflictsnorconfusion

caneverarise.

93

Quantitativeanalysis

WeconsiderKcardsandn=5cellsofequalmachine

time(therateoftransitionsoutM

iis4.0)

Firstcase:Inputandoutputinventoryinthecells

Inputbufferinventory

Outputbufferinventory

Cell1Card

2Cards3Cards1Card

2Cards3Cards

10.486

1.041

1.474

0.514

0.958

1.526

20.486

1.040

1.470

0.383

0.713

1.131

30.486

1.047

1.478

0.282

0.524

0.811

40.486

1.056

1.490

0.170

0.316

0.472

50.486

1.073

1.515

0.000

0.000

0.000

Theinputinventoryisfairlyconstant,whiletheoutput

inventorydecreasesasthecellpositionincreases.

94

263

Secondcase:faultfreeversusfailurepronesystems

idleM

i

OB

ibusyM

iIB

i

BB

i

OK

iFAILED

i

repair

i

failure

i

outM

iexitCell i

inM

ienterCell i

Ki

ModelofaKanbancellthatcanfail

95

•Cellscanfailindependently;

•Failurerateis0.02;

•Repairrateis0.4.

1.60

1.80

2.00

2.20

2.40

2.60

2.80

3.00

12

3

Failure

free system

Failure

prone system

Throughput

K

Failurefreevs.“all

cellscanfail”

K

0.03

0

0.03

2

0.03

4

0.03

6

0.03

8

0.04

0

12

34

56

Relative Throughput

Reduction

Failure in

middle cell

Failure in

last cell

0.02

8

Middlecellfailurevs.

finalcellfailure

InaperfectlybalancedKanbansystemthecellperfor-

manceisposition-dependent. 96

264

CONCLUSIONS

97

StochasticPetrinettechniquesareattractivebecause

theyprovideaperformanceevaluationapproachbased

onaformaldescription.

Thisallowstheuseofthesamelanguageforthe

•specification

•validation

•performanceevaluation

•implementation

•documentation

ofasystem.

98

265

Twoarethemaindirectionsoftheresearchbeingpresently

conductedinthefieldofGSPN-basedperformanceeval-

uation.

1.ExtensionsoftheGSPNanalysisapproachtoenvi-

ronmentsinwhichtokenspossessanidentityhasal-

readybeenproposedbyseveralauthors,andmore

workisbeingperformedtoobtainanenvironment

withahighdescriptivepowerinwhichthemodel

specificationissimple.

2.Variousapproachesarebeingpursuedforthereduc-

tionofthecomplexityofthesolutioncomputation

withstochastictechniques,possiblyproducingonly

partialorapproximateresults.

SuccessesinthesetwofieldswouldmakeGSPNapromi-

nentmodelingtechniqueinthewholeareaofdistributed

systems.

99

266

Documents

Petri Nets 2000 · Petri Nets 2000 21st International Conference on Application and Theory of Petri Nets Aarhus, Denmark, June 26-30, 2000 Introductory Tutorial Petri Nets